What is the general solution for 1+sin^2(x)+cos^4(x)=0 ?

The general solution for 1+sin^2(x)+cos^4(x)=0 is No Solution for x\in\mathbb{R}

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1+sin^2(x)+cos^4(x)=0

Solution

1 + sin^{2} (x) + cos^{4} (x) = 0

Solution

No Solution for x \in R

Solution steps

1 + sin^{2} (x) + cos^{4} (x) = 0

Rewrite using trig identities

1 + cos^{4} (x) + sin^{2} (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= 1 + cos^{4} (x) + 1 - cos^{2} (x)

Simplify

1 + cos^{4} (x) + 1 - cos^{2} (x) : cos^{4} (x) - cos^{2} (x) + 2

1 + cos^{4} (x) + 1 - cos^{2} (x)

Group like terms

= cos^{4} (x) - cos^{2} (x) + 1 + 1

Add the numbers:

1 + 1 = 2

= cos^{4} (x) - cos^{2} (x) + 2

= cos^{4} (x) - cos^{2} (x) + 2

2 - cos^{2} (x) + cos^{4} (x) = 0

Solve by substitution

2 - cos^{2} (x) + cos^{4} (x) = 0

Let:

cos (x) = u

2 - u^{2} + u^{4} = 0

2 - u^{2} + u^{4} = 0 : u = \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i, u = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}} - \frac{2 2 - 1}{2} i, u = - \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i, u = - \frac{7}{2 ( - 2 2 - 1 )} - \frac{2 2 - 1}{2} i

2 - u^{2} + u^{4} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{4} - u^{2} + 2 = 0

Rewrite the equation with

a = u^{2}

and

a^{2} = u^{4}

a^{2} - a + 2 = 0

Solve

a^{2} - a + 2 = 0 : a = \frac{1}{2} + i \frac{7}{2}, a = \frac{1}{2} - i \frac{7}{2}

a^{2} - a + 2 = 0

Solve with the quadratic formula

a^{2} - a + 2 = 0

Quadratic Equation Formula:

For

a = 1, b = - 1, c = 2

a_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 1 \cdot 2}{2 \cdot 1}

a_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 1 \cdot 2}{2 \cdot 1}

Simplify

(- 1)^{2} - 4 \cdot 1 \cdot 2 : 7 i

(- 1)^{2} - 4 \cdot 1 \cdot 2

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 1 \cdot 2 = 8

4 \cdot 1 \cdot 2

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 8

= 1 - 8

Subtract the numbers:

1 - 8 = - 7

= - 7

Apply radical rule:

- a = - 1 a

- 7 = - 1 7

= - 1 7

Apply imaginary number rule:

- 1 = i

= 7 i

a_{1, 2} = \frac{- ( - 1 ) \pm 7 i}{2 \cdot 1}

Separate the solutions

a_{1} = \frac{- ( - 1 ) + 7 i}{2 \cdot 1}, a_{2} = \frac{- ( - 1 ) - 7 i}{2 \cdot 1}

a = \frac{- ( - 1 ) + 7 i}{2 \cdot 1} : \frac{1}{2} + i \frac{7}{2}

\frac{- ( - 1 ) + 7 i}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{1 + 7 i}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{1 + 7 i}{2}

Rewrite

\frac{1 + 7 i}{2}

in standard complex form:

\frac{1}{2} + \frac{7}{2} i

\frac{1 + 7 i}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{1 + 7 i}{2} = \frac{1}{2} + \frac{7 i}{2}

= \frac{1}{2} + \frac{7 i}{2}

= \frac{1}{2} + \frac{7}{2} i

a = \frac{- ( - 1 ) - 7 i}{2 \cdot 1} : \frac{1}{2} - i \frac{7}{2}

\frac{- ( - 1 ) - 7 i}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{1 - 7 i}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{1 - 7 i}{2}

Rewrite

\frac{1 - 7 i}{2}

in standard complex form:

\frac{1}{2} - \frac{7}{2} i

\frac{1 - 7 i}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{1 - 7 i}{2} = \frac{1}{2} - \frac{7 i}{2}

= \frac{1}{2} - \frac{7 i}{2}

= \frac{1}{2} - \frac{7}{2} i

The solutions to the quadratic equation are:

a = \frac{1}{2} + i \frac{7}{2}, a = \frac{1}{2} - i \frac{7}{2}

a = \frac{1}{2} + i \frac{7}{2}, a = \frac{1}{2} - i \frac{7}{2}

Substitute back

a = u^{2},

solve for

u

Solve

u^{2} = \frac{1}{2} + i \frac{7}{2} : u = \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i, u = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}} - \frac{2 2 - 1}{2} i

u^{2} = \frac{1}{2} + i \frac{7}{2}

Substitute

u = a + bi

(a + bi)^{2} = \frac{1}{2} + i \frac{7}{2}

Expand

(a + bi)^{2} : (a^{2} - b^{2}) + 2 iab

(a + bi)^{2}

= (a + ib)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = a, b = bi

= a^{2} + 2 abi + (bi)^{2}

(bi)^{2} = - b^{2}

(bi)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= i^{2} b^{2}

i^{2} = - 1

i^{2}

Apply imaginary number rule:

i^{2} = - 1

= - 1

= (- 1) b^{2}

Refine

= - b^{2}

= a^{2} + 2 iab - b^{2}

Rewrite

a^{2} + 2 iab - b^{2}

in standard complex form:

(a^{2} - b^{2}) + 2 abi

a^{2} + 2 iab - b^{2}

Group the real part and the imaginary part of the complex number

= (a^{2} - b^{2}) + 2 abi

= (a^{2} - b^{2}) + 2 abi

(a^{2} - b^{2}) + 2 iab = \frac{1}{2} + i \frac{7}{2}

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[a^{2} - b^{2} = \frac{1}{2} 2 ab = \frac{7}{2}]

[a^{2} - b^{2} = \frac{1}{2} 2 ab = \frac{7}{2}] : a = \frac{7}{2 2 2 - 1}, a = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}, b = \frac{2 2 - 1}{2} b = - \frac{2 2 - 1}{2}

[a^{2} - b^{2} = \frac{1}{2} 2 ab = \frac{7}{2}]

Isolate

a

for

2 ab = \frac{7}{2} : a = \frac{7}{4 b}

2 ab = \frac{7}{2}

Divide both sides by

2 b

2 ab = \frac{7}{2}

Divide both sides by

2 b

\frac{2 ab}{2 b} = \frac{\frac{7}{2}}{2 b}

Simplify

\frac{2 ab}{2 b} = \frac{\frac{7}{2}}{2 b}

Simplify

\frac{2 ab}{2 b} : a

\frac{2 ab}{2 b}

Divide the numbers:

\frac{2}{2} = 1

= \frac{ab}{b}

Cancel the common factor:

b

= a

Simplify

\frac{\frac{7}{2}}{2 b} : \frac{7}{4 b}

\frac{\frac{7}{2}}{2 b}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{7}{2 \cdot 2 b}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{7}{4 b}

a = \frac{7}{4 b}

a = \frac{7}{4 b}

a = \frac{7}{4 b}

Plug the solutions

a = \frac{7}{4 b}

into

a^{2} - b^{2} = \frac{1}{2}

For

a^{2} - b^{2} = \frac{1}{2}

, subsitute

a

with

\frac{7}{4 b} : b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

For

a^{2} - b^{2} = \frac{1}{2}

, subsitute

a

with

\frac{7}{4 b}

(\frac{7}{4 b})^{2} - b^{2} = \frac{1}{2}

Solve

(\frac{7}{4 b})^{2} - b^{2} = \frac{1}{2} : b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

(\frac{7}{4 b})^{2} - b^{2} = \frac{1}{2}

Multiply by LCM

(\frac{7}{4 b})^{2} - b^{2} = \frac{1}{2}

Simplify

(\frac{7}{4 b})^{2} : \frac{7}{16 b ^{2}}

(\frac{7}{4 b})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 7 ) ^{2}}{( 4 b ) ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(4 b)^{2} = 4^{2} b^{2}

= \frac{( 7 ) ^{2}}{4 ^{2} b ^{2}}

(7)^{2} : 7

Apply radical rule:

a = a^{\frac{1}{2}}

= (7^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 7^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 7

= \frac{7}{4 ^{2} b ^{2}}

4^{2} = 16

= \frac{7}{16 b ^{2}}

\frac{7}{16 b ^{2}} - b^{2} = \frac{1}{2}

Find Least Common Multiplier of

16 b^{2}, 2 : 16 b^{2}

16 b^{2}, 2

Lowest Common Multiplier (LCM)

Least Common Multiplier of

16, 2 : 16

16, 2

Least Common Multiplier (LCM)

Prime factorization of

16 : 2 \cdot 2 \cdot 2 \cdot 2

16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

16

2

= 2 \cdot 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 \cdot 2 = 16

= 16

Compute an expression comprised of factors that appear either in

16 b^{2}

2

= 16 b^{2}

Multiply by LCM=

16 b^{2}

\frac{7}{16 b ^{2}} \cdot 16 b^{2} - b^{2} \cdot 16 b^{2} = \frac{1}{2} \cdot 16 b^{2}

Simplify

\frac{7}{16 b ^{2}} \cdot 16 b^{2} - b^{2} \cdot 16 b^{2} = \frac{1}{2} \cdot 16 b^{2}

Simplify

\frac{7}{16 b ^{2}} \cdot 16 b^{2} : 7

\frac{7}{16 b ^{2}} \cdot 16 b^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{7 \cdot 16 b ^{2}}{16 b ^{2}}

Cancel the common factor:

16

= \frac{7 b ^{2}}{b ^{2}}

Cancel the common factor:

b^{2}

= 7

Simplify

- b^{2} \cdot 16 b^{2} : - 16 b^{4}

- b^{2} \cdot 16 b^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

b^{2} b^{2} = b^{2 + 2}

= - 16 b^{2 + 2}

Add the numbers:

2 + 2 = 4

= - 16 b^{4}

Simplify

\frac{1}{2} \cdot 16 b^{2} : 8 b^{2}

\frac{1}{2} \cdot 16 b^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 16}{2} b^{2}

\frac{1 \cdot 16}{2} = 8

\frac{1 \cdot 16}{2}

Multiply the numbers:

1 \cdot 16 = 16

= \frac{16}{2}

Divide the numbers:

\frac{16}{2} = 8

= 8

= 8 b^{2}

7 - 16 b^{4} = 8 b^{2}

7 - 16 b^{4} = 8 b^{2}

7 - 16 b^{4} = 8 b^{2}

Solve

7 - 16 b^{4} = 8 b^{2} : b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

7 - 16 b^{4} = 8 b^{2}

Move

8 b^{2}

to the left side

7 - 16 b^{4} = 8 b^{2}

Subtract

8 b^{2}

from both sides

7 - 16 b^{4} - 8 b^{2} = 8 b^{2} - 8 b^{2}

Simplify

7 - 16 b^{4} - 8 b^{2} = 0

7 - 16 b^{4} - 8 b^{2} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 16 b^{4} - 8 b^{2} + 7 = 0

Rewrite the equation with

u = b^{2}

and

u^{2} = b^{4}

- 16 u^{2} - 8 u + 7 = 0

Solve

- 16 u^{2} - 8 u + 7 = 0 : u = - \frac{1 + 2 2}{4}, u = \frac{2 2 - 1}{4}

- 16 u^{2} - 8 u + 7 = 0

Solve with the quadratic formula

- 16 u^{2} - 8 u + 7 = 0

Quadratic Equation Formula:

For

a = - 16, b = - 8, c = 7

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 ( - 16 ) \cdot 7}{2 ( - 16 )}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 ( - 16 ) \cdot 7}{2 ( - 16 )}

(- 8)^{2} - 4 (- 16) \cdot 7 = 162

(- 8)^{2} - 4 (- 16) \cdot 7

Apply rule

- (- a) = a

= (- 8)^{2} + 4 \cdot 16 \cdot 7

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 8)^{2} = 8^{2}

= 8^{2} + 4 \cdot 16 \cdot 7

Multiply the numbers:

4 \cdot 16 \cdot 7 = 448

= 8^{2} + 448

8^{2} = 64

= 64 + 448

Add the numbers:

64 + 448 = 512

= 512

Prime factorization of

512 : 2^{9}

512

512

divides by

2512 = 256 \cdot 2

= 2 \cdot 256

256

divides by

2256 = 128 \cdot 2

= 2 \cdot 2 \cdot 128

128

divides by

2128 = 64 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 64

64

divides by

264 = 32 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

= 2^{9}

= 2^{9}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{8} \cdot 2

Apply radical rule:

= 2 2^{8}

Apply radical rule:

2^{8} = 2^{\frac{8}{2}} = 2^{4}

= 2^{4} 2

Refine

= 162

u_{1, 2} = \frac{- ( - 8 ) \pm 16 2}{2 ( - 16 )}

Separate the solutions

u_{1} = \frac{- ( - 8 ) + 16 2}{2 ( - 16 )}, u_{2} = \frac{- ( - 8 ) - 16 2}{2 ( - 16 )}

u = \frac{- ( - 8 ) + 16 2}{2 ( - 16 )} : - \frac{1 + 2 2}{4}

\frac{- ( - 8 ) + 16 2}{2 ( - 16 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{8 + 16 2}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{8 + 16 2}{- 32}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{8 + 16 2}{32}

Cancel

\frac{8 + 16 2}{32} : \frac{1 + 2 2}{4}

\frac{8 + 16 2}{32}

Factor

8 + 162 : 8 (1 + 22)

8 + 162

Rewrite as

= 8 \cdot 1 + 8 \cdot 22

Factor out common term

8

= 8 (1 + 22)

= \frac{8 ( 1 + 2 2 )}{32}

Cancel the common factor:

8

= \frac{1 + 2 2}{4}

= - \frac{1 + 2 2}{4}

u = \frac{- ( - 8 ) - 16 2}{2 ( - 16 )} : \frac{2 2 - 1}{4}

\frac{- ( - 8 ) - 16 2}{2 ( - 16 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{8 - 16 2}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{8 - 16 2}{- 32}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

8 - 162 = - (162 - 8)

= \frac{16 2 - 8}{32}

Factor

162 - 8 : 8 (22 - 1)

162 - 8

Rewrite as

= 8 \cdot 22 - 8 \cdot 1

Factor out common term

8

= 8 (22 - 1)

= \frac{8 ( 2 2 - 1 )}{32}

Cancel the common factor:

8

= \frac{2 2 - 1}{4}

The solutions to the quadratic equation are:

u = - \frac{1 + 2 2}{4}, u = \frac{2 2 - 1}{4}

u = - \frac{1 + 2 2}{4}, u = \frac{2 2 - 1}{4}

Substitute back

u = b^{2},

solve for

b

Solve

b^{2} = - \frac{1 + 2 2}{4} :

No Solution for

b \in R

b^{2} = - \frac{1 + 2 2}{4}

x^{2}

cannot be negative for

x \in R

No Solution for b \in R

Solve

b^{2} = \frac{2 2 - 1}{4} : b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

b^{2} = \frac{2 2 - 1}{4}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

b = \frac{2 2 - 1}{4}, b = - \frac{2 2 - 1}{4}

\frac{2 2 - 1}{4} = \frac{2 2 - 1}{2}

\frac{2 2 - 1}{4}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{2 2 - 1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

a^{2} = a, a \geq 0

2^{2} = 2

= 2

= \frac{2 2 - 1}{2}

- \frac{2 2 - 1}{4} = - \frac{2 2 - 1}{2}

- \frac{2 2 - 1}{4}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{2 2 - 1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

a^{2} = a, a \geq 0

2^{2} = 2

= 2

= - \frac{2 2 - 1}{2}

b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

The solutions are

b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

Verify Solutions

Find undefined (singularity) points:

b = 0

Take the denominator(s) of

(\frac{7}{4 b})^{2} - b^{2}

and compare to zero

Solve

4 b = 0 : b = 0

4 b = 0

Divide both sides by

4

4 b = 0

Divide both sides by

4

\frac{4 b}{4} = \frac{0}{4}

Simplify

b = 0

b = 0

The following points are undefined

b = 0

Combine undefined points with solutions:

b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

Plug the solutions

b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

into

2 ab = \frac{7}{2}

For

2 ab = \frac{7}{2}

, subsitute

b

with

\frac{2 2 - 1}{2} : a = \frac{7}{2 2 2 - 1}

For

2 ab = \frac{7}{2}

, subsitute

b

with

\frac{2 2 - 1}{2}

2 a \frac{2 2 - 1}{2} = \frac{7}{2}

Solve

2 a \frac{2 2 - 1}{2} = \frac{7}{2} : a = \frac{7}{2 2 2 - 1}

2 a \frac{2 2 - 1}{2} = \frac{7}{2}

Multiply both sides by

2

2 a \frac{2 2 - 1}{2} = \frac{7}{2}

Multiply both sides by

2

2 \cdot 2 a \frac{2 2 - 1}{2} = \frac{2 7}{2}

Simplify

2 a 22 - 1 = 7

2 a 22 - 1 = 7

Divide both sides by

2 22 - 1

2 a 22 - 1 = 7

Divide both sides by

2 22 - 1

\frac{2 a 2 2 - 1}{2 2 2 - 1} = \frac{7}{2 2 2 - 1}

Simplify

a = \frac{7}{2 2 2 - 1}

a = \frac{7}{2 2 2 - 1}

For

2 ab = \frac{7}{2}

, subsitute

b

with

- \frac{2 2 - 1}{2} : a = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}

For

2 ab = \frac{7}{2}

, subsitute

b

with

- \frac{2 2 - 1}{2}

2 a (- \frac{2 2 - 1}{2}) = \frac{7}{2}

Solve

2 a (- \frac{2 2 - 1}{2}) = \frac{7}{2} : a = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}

2 a (- \frac{2 2 - 1}{2}) = \frac{7}{2}

Divide both sides by

2 (- \frac{2 2 - 1}{2})

2 a (- \frac{2 2 - 1}{2}) = \frac{7}{2}

Divide both sides by

2 (- \frac{2 2 - 1}{2})

\frac{2 a ( - \frac{2 2 - 1}{2} )}{2 ( - \frac{2 2 - 1}{2} )} = \frac{\frac{7}{2}}{2 ( - \frac{2 2 - 1}{2} )}

Simplify

\frac{2 a ( - \frac{2 2 - 1}{2} )}{2 ( - \frac{2 2 - 1}{2} )} = \frac{\frac{7}{2}}{2 ( - \frac{2 2 - 1}{2} )}

Simplify

\frac{2 a ( - \frac{2 2 - 1}{2} )}{2 ( - \frac{2 2 - 1}{2} )} : a

\frac{2 a ( - \frac{2 2 - 1}{2} )}{2 ( - \frac{2 2 - 1}{2} )}

Simplify

\frac{2 a ( - \frac{2 2 - 1}{2} )}{2 ( - \frac{2 2 - 1}{2} )} : \frac{- 2 a \frac{2 2 - 1}{2}}{- 2 \cdot \frac{2 2 - 1}{2}}

\frac{2 a ( - \frac{2 2 - 1}{2} )}{2 ( - \frac{2 2 - 1}{2} )}

Apply rule:

a (- b) = - ab

2 a (- \frac{2 2 - 1}{2}) = - 2 a \frac{2 2 - 1}{2}

= \frac{- 2 a \frac{2 2 - 1}{2}}{2 ( - \frac{2 2 - 1}{2} )}

Apply rule:

a (- b) = - ab

2 (- \frac{2 2 - 1}{2}) = - 2 \cdot \frac{2 2 - 1}{2}

= \frac{- 2 a \frac{2 2 - 1}{2}}{- 2 \cdot \frac{2 2 - 1}{2}}

= \frac{- 2 a \frac{2 2 - 1}{2}}{- 2 \cdot \frac{2 2 - 1}{2}}

Cancel the common factor:

- 2

= \frac{a \frac{2 2 - 1}{2}}{\frac{2 2 - 1}{2}}

Cancel the common factor:

\frac{2 2 - 1}{2}

= a

Simplify

\frac{\frac{7}{2}}{2 ( - \frac{2 2 - 1}{2} )} : - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}

\frac{\frac{7}{2}}{2 ( - \frac{2 2 - 1}{2} )}

Apply the fraction rule:

\frac{\frac{a}{b}}{c} = \frac{a}{b \cdot c}

= \frac{7}{2 \cdot 2 ( - \frac{2 2 - 1}{2} )}

Apply rule:

a (- b) = - ab

2 \cdot 2 (- \frac{2 2 - 1}{2}) = - 2 \cdot 2 \cdot \frac{2 2 - 1}{2}

= \frac{7}{- 2 \cdot 2 \cdot \frac{2 2 - 1}{2}}

2 \cdot 2 = 2^{2}

2 \cdot 2

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2 \cdot 2 = 2^{1 + 1}

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

= \frac{7}{- 2 ^{2} \cdot \frac{2 2 - 1}{2}}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}

a = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}

a = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}

a = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}

Verify solutions by plugging them into the original equations

Check the solutions by plugging them into

a^{2} - b^{2} = \frac{1}{2}

Remove the ones that don't agree with the equation.

Check the solution

a = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}, b = - \frac{2 2 - 1}{2} :

True

a^{2} - b^{2} = \frac{1}{2}

Plug in

a = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}, b = - \frac{2 2 - 1}{2}

- \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}^{2} - (- \frac{2 2 - 1}{2})^{2} = \frac{1}{2}

Refine

\frac{1}{2} = \frac{1}{2}

True

Check the solution

a = \frac{7}{2 2 2 - 1}, b = \frac{2 2 - 1}{2} :

True

a^{2} - b^{2} = \frac{1}{2}

Plug in

a = \frac{7}{2 2 2 - 1}, b = \frac{2 2 - 1}{2}

(\frac{7}{2 2 2 - 1})^{2} - (\frac{2 2 - 1}{2})^{2} = \frac{1}{2}

Refine

\frac{1}{2} = \frac{1}{2}

True

Check the solutions by plugging them into

2 ab = \frac{7}{2}

Remove the ones that don't agree with the equation.

Check the solution

a = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}, b = - \frac{2 2 - 1}{2} :

True

2 ab = \frac{7}{2}

Plug in

a = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}, b = - \frac{2 2 - 1}{2}

2 - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}} (- \frac{2 2 - 1}{2}) = \frac{7}{2}

Refine

\frac{7}{2} = \frac{7}{2}

True

Check the solution

a = \frac{7}{2 2 2 - 1}, b = \frac{2 2 - 1}{2} :

True

2 ab = \frac{7}{2}

Plug in

a = \frac{7}{2 2 2 - 1}, b = \frac{2 2 - 1}{2}

2 \cdot \frac{7}{2 2 2 - 1} \cdot \frac{2 2 - 1}{2} = \frac{7}{2}

Refine

\frac{7}{2} = \frac{7}{2}

True

Therefore, the final solutions for

a^{2} - b^{2} = \frac{1}{2}, 2 ab = \frac{7}{2}

are

a = \frac{7}{2 2 2 - 1}, a = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}}, b = \frac{2 2 - 1}{2} b = - \frac{2 2 - 1}{2}

Substitute back

u = a + bi

u = \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i, u = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}} - \frac{2 2 - 1}{2} i

Solve

u^{2} = \frac{1}{2} - i \frac{7}{2} : u = - \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i, u = - \frac{7}{2 ( - 2 2 - 1 )} - \frac{2 2 - 1}{2} i

u^{2} = \frac{1}{2} - i \frac{7}{2}

Substitute

u = a + bi

(a + bi)^{2} = \frac{1}{2} - i \frac{7}{2}

Expand

(a + bi)^{2} : (a^{2} - b^{2}) + 2 iab

(a + bi)^{2}

= (a + ib)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = a, b = bi

= a^{2} + 2 abi + (bi)^{2}

(bi)^{2} = - b^{2}

(bi)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= i^{2} b^{2}

i^{2} = - 1

i^{2}

Apply imaginary number rule:

i^{2} = - 1

= - 1

= (- 1) b^{2}

Refine

= - b^{2}

= a^{2} + 2 iab - b^{2}

Rewrite

a^{2} + 2 iab - b^{2}

in standard complex form:

(a^{2} - b^{2}) + 2 abi

a^{2} + 2 iab - b^{2}

Group the real part and the imaginary part of the complex number

= (a^{2} - b^{2}) + 2 abi

= (a^{2} - b^{2}) + 2 abi

(a^{2} - b^{2}) + 2 iab = \frac{1}{2} - i \frac{7}{2}

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[a^{2} - b^{2} = \frac{1}{2} 2 ab = - \frac{7}{2}]

[a^{2} - b^{2} = \frac{1}{2} 2 ab = - \frac{7}{2}] : a = - \frac{7}{2 2 2 - 1}, a = - \frac{7}{2 ( - 2 2 - 1 )}, b = \frac{2 2 - 1}{2} b = - \frac{2 2 - 1}{2}

[a^{2} - b^{2} = \frac{1}{2} 2 ab = - \frac{7}{2}]

Isolate

a

for

2 ab = - \frac{7}{2} : a = - \frac{7}{4 b}

2 ab = - \frac{7}{2}

Divide both sides by

2 b

2 ab = - \frac{7}{2}

Divide both sides by

2 b

\frac{2 ab}{2 b} = \frac{- \frac{7}{2}}{2 b}

Simplify

\frac{2 ab}{2 b} = \frac{- \frac{7}{2}}{2 b}

Simplify

\frac{2 ab}{2 b} : a

\frac{2 ab}{2 b}

Divide the numbers:

\frac{2}{2} = 1

= \frac{ab}{b}

Cancel the common factor:

b

= a

Simplify

\frac{- \frac{7}{2}}{2 b} : - \frac{7}{4 b}

\frac{- \frac{7}{2}}{2 b}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{7}{2}}{2 b}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

\frac{\frac{7}{2}}{2 b} = \frac{7}{2 \cdot 2 b}

= - \frac{7}{2 \cdot 2 b}

Multiply the numbers:

2 \cdot 2 = 4

= - \frac{7}{4 b}

a = - \frac{7}{4 b}

a = - \frac{7}{4 b}

a = - \frac{7}{4 b}

Plug the solutions

a = - \frac{7}{4 b}

into

a^{2} - b^{2} = \frac{1}{2}

For

a^{2} - b^{2} = \frac{1}{2}

, subsitute

a

with

- \frac{7}{4 b} : b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

For

a^{2} - b^{2} = \frac{1}{2}

, subsitute

a

with

- \frac{7}{4 b}

(- \frac{7}{4 b})^{2} - b^{2} = \frac{1}{2}

Solve

(- \frac{7}{4 b})^{2} - b^{2} = \frac{1}{2} : b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

(- \frac{7}{4 b})^{2} - b^{2} = \frac{1}{2}

Multiply by LCM

(- \frac{7}{4 b})^{2} - b^{2} = \frac{1}{2}

Simplify

(- \frac{7}{4 b})^{2} : \frac{7}{16 b ^{2}}

(- \frac{7}{4 b})^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- \frac{7}{4 b})^{2} = (\frac{7}{4 b})^{2}

= (\frac{7}{4 b})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 7 ) ^{2}}{( 4 b ) ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(4 b)^{2} = 4^{2} b^{2}

= \frac{( 7 ) ^{2}}{4 ^{2} b ^{2}}

(7)^{2} : 7

Apply radical rule:

a = a^{\frac{1}{2}}

= (7^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 7^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 7

= \frac{7}{4 ^{2} b ^{2}}

4^{2} = 16

= \frac{7}{16 b ^{2}}

\frac{7}{16 b ^{2}} - b^{2} = \frac{1}{2}

Find Least Common Multiplier of

16 b^{2}, 2 : 16 b^{2}

16 b^{2}, 2

Lowest Common Multiplier (LCM)

Least Common Multiplier of

16, 2 : 16

16, 2

Least Common Multiplier (LCM)

Prime factorization of

16 : 2 \cdot 2 \cdot 2 \cdot 2

16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

16

2

= 2 \cdot 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 \cdot 2 = 16

= 16

Compute an expression comprised of factors that appear either in

16 b^{2}

2

= 16 b^{2}

Multiply by LCM=

16 b^{2}

\frac{7}{16 b ^{2}} \cdot 16 b^{2} - b^{2} \cdot 16 b^{2} = \frac{1}{2} \cdot 16 b^{2}

Simplify

\frac{7}{16 b ^{2}} \cdot 16 b^{2} - b^{2} \cdot 16 b^{2} = \frac{1}{2} \cdot 16 b^{2}

Simplify

\frac{7}{16 b ^{2}} \cdot 16 b^{2} : 7

\frac{7}{16 b ^{2}} \cdot 16 b^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{7 \cdot 16 b ^{2}}{16 b ^{2}}

Cancel the common factor:

16

= \frac{7 b ^{2}}{b ^{2}}

Cancel the common factor:

b^{2}

= 7

Simplify

- b^{2} \cdot 16 b^{2} : - 16 b^{4}

- b^{2} \cdot 16 b^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

b^{2} b^{2} = b^{2 + 2}

= - 16 b^{2 + 2}

Add the numbers:

2 + 2 = 4

= - 16 b^{4}

Simplify

\frac{1}{2} \cdot 16 b^{2} : 8 b^{2}

\frac{1}{2} \cdot 16 b^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 16}{2} b^{2}

\frac{1 \cdot 16}{2} = 8

\frac{1 \cdot 16}{2}

Multiply the numbers:

1 \cdot 16 = 16

= \frac{16}{2}

Divide the numbers:

\frac{16}{2} = 8

= 8

= 8 b^{2}

7 - 16 b^{4} = 8 b^{2}

7 - 16 b^{4} = 8 b^{2}

7 - 16 b^{4} = 8 b^{2}

Solve

7 - 16 b^{4} = 8 b^{2} : b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

7 - 16 b^{4} = 8 b^{2}

Move

8 b^{2}

to the left side

7 - 16 b^{4} = 8 b^{2}

Subtract

8 b^{2}

from both sides

7 - 16 b^{4} - 8 b^{2} = 8 b^{2} - 8 b^{2}

Simplify

7 - 16 b^{4} - 8 b^{2} = 0

7 - 16 b^{4} - 8 b^{2} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 16 b^{4} - 8 b^{2} + 7 = 0

Rewrite the equation with

u = b^{2}

and

u^{2} = b^{4}

- 16 u^{2} - 8 u + 7 = 0

Solve

- 16 u^{2} - 8 u + 7 = 0 : u = - \frac{1 + 2 2}{4}, u = \frac{2 2 - 1}{4}

- 16 u^{2} - 8 u + 7 = 0

Solve with the quadratic formula

- 16 u^{2} - 8 u + 7 = 0

Quadratic Equation Formula:

For

a = - 16, b = - 8, c = 7

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 ( - 16 ) \cdot 7}{2 ( - 16 )}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 ( - 16 ) \cdot 7}{2 ( - 16 )}

(- 8)^{2} - 4 (- 16) \cdot 7 = 162

(- 8)^{2} - 4 (- 16) \cdot 7

Apply rule

- (- a) = a

= (- 8)^{2} + 4 \cdot 16 \cdot 7

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 8)^{2} = 8^{2}

= 8^{2} + 4 \cdot 16 \cdot 7

Multiply the numbers:

4 \cdot 16 \cdot 7 = 448

= 8^{2} + 448

8^{2} = 64

= 64 + 448

Add the numbers:

64 + 448 = 512

= 512

Prime factorization of

512 : 2^{9}

512

512

divides by

2512 = 256 \cdot 2

= 2 \cdot 256

256

divides by

2256 = 128 \cdot 2

= 2 \cdot 2 \cdot 128

128

divides by

2128 = 64 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 64

64

divides by

264 = 32 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

= 2^{9}

= 2^{9}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{8} \cdot 2

Apply radical rule:

= 2 2^{8}

Apply radical rule:

2^{8} = 2^{\frac{8}{2}} = 2^{4}

= 2^{4} 2

Refine

= 162

u_{1, 2} = \frac{- ( - 8 ) \pm 16 2}{2 ( - 16 )}

Separate the solutions

u_{1} = \frac{- ( - 8 ) + 16 2}{2 ( - 16 )}, u_{2} = \frac{- ( - 8 ) - 16 2}{2 ( - 16 )}

u = \frac{- ( - 8 ) + 16 2}{2 ( - 16 )} : - \frac{1 + 2 2}{4}

\frac{- ( - 8 ) + 16 2}{2 ( - 16 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{8 + 16 2}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{8 + 16 2}{- 32}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{8 + 16 2}{32}

Cancel

\frac{8 + 16 2}{32} : \frac{1 + 2 2}{4}

\frac{8 + 16 2}{32}

Factor

8 + 162 : 8 (1 + 22)

8 + 162

Rewrite as

= 8 \cdot 1 + 8 \cdot 22

Factor out common term

8

= 8 (1 + 22)

= \frac{8 ( 1 + 2 2 )}{32}

Cancel the common factor:

8

= \frac{1 + 2 2}{4}

= - \frac{1 + 2 2}{4}

u = \frac{- ( - 8 ) - 16 2}{2 ( - 16 )} : \frac{2 2 - 1}{4}

\frac{- ( - 8 ) - 16 2}{2 ( - 16 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{8 - 16 2}{- 2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{8 - 16 2}{- 32}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

8 - 162 = - (162 - 8)

= \frac{16 2 - 8}{32}

Factor

162 - 8 : 8 (22 - 1)

162 - 8

Rewrite as

= 8 \cdot 22 - 8 \cdot 1

Factor out common term

8

= 8 (22 - 1)

= \frac{8 ( 2 2 - 1 )}{32}

Cancel the common factor:

8

= \frac{2 2 - 1}{4}

The solutions to the quadratic equation are:

u = - \frac{1 + 2 2}{4}, u = \frac{2 2 - 1}{4}

u = - \frac{1 + 2 2}{4}, u = \frac{2 2 - 1}{4}

Substitute back

u = b^{2},

solve for

b

Solve

b^{2} = - \frac{1 + 2 2}{4} :

No Solution for

b \in R

b^{2} = - \frac{1 + 2 2}{4}

x^{2}

cannot be negative for

x \in R

No Solution for b \in R

Solve

b^{2} = \frac{2 2 - 1}{4} : b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

b^{2} = \frac{2 2 - 1}{4}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

b = \frac{2 2 - 1}{4}, b = - \frac{2 2 - 1}{4}

\frac{2 2 - 1}{4} = \frac{2 2 - 1}{2}

\frac{2 2 - 1}{4}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{2 2 - 1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

a^{2} = a, a \geq 0

2^{2} = 2

= 2

= \frac{2 2 - 1}{2}

- \frac{2 2 - 1}{4} = - \frac{2 2 - 1}{2}

- \frac{2 2 - 1}{4}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{2 2 - 1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

a^{2} = a, a \geq 0

2^{2} = 2

= 2

= - \frac{2 2 - 1}{2}

b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

The solutions are

b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

Verify Solutions

Find undefined (singularity) points:

b = 0

Take the denominator(s) of

(- \frac{7}{4 b})^{2} - b^{2}

and compare to zero

Solve

4 b = 0 : b = 0

4 b = 0

Divide both sides by

4

4 b = 0

Divide both sides by

4

\frac{4 b}{4} = \frac{0}{4}

Simplify

b = 0

b = 0

The following points are undefined

b = 0

Combine undefined points with solutions:

b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

Plug the solutions

b = \frac{2 2 - 1}{2}, b = - \frac{2 2 - 1}{2}

into

2 ab = - \frac{7}{2}

For

2 ab = - \frac{7}{2}

, subsitute

b

with

\frac{2 2 - 1}{2} : a = - \frac{7}{2 2 2 - 1}

For

2 ab = - \frac{7}{2}

, subsitute

b

with

\frac{2 2 - 1}{2}

2 a \frac{2 2 - 1}{2} = - \frac{7}{2}

Solve

2 a \frac{2 2 - 1}{2} = - \frac{7}{2} : a = - \frac{7}{2 2 2 - 1}

2 a \frac{2 2 - 1}{2} = - \frac{7}{2}

Multiply both sides by

2

2 a \frac{2 2 - 1}{2} = - \frac{7}{2}

Multiply both sides by

2

2 \cdot 2 a \frac{2 2 - 1}{2} = 2 (- \frac{7}{2})

Simplify

2 \cdot 2 a \frac{2 2 - 1}{2} = 2 (- \frac{7}{2})

Simplify

2 \cdot 2 a \frac{2 2 - 1}{2} : 2 a 22 - 1

2 \cdot 2 a \frac{2 2 - 1}{2}

2 \cdot 2 = 2^{2}

2 \cdot 2

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2 \cdot 2 = 2^{1 + 1}

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

= 2^{2} a \frac{2 2 - 1}{2}

Apply the fraction rule:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 ^{2} a 2 2 - 1}{2}

Cancel

\frac{2 ^{2} a 2 2 - 1}{2} : 2 a 22 - 1

\frac{2 ^{2} a 2 2 - 1}{2}

\frac{2 ^{2}}{2} = 2

\frac{2 ^{2}}{2}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

2^{2} = 2 \cdot 2

= \frac{2 \cdot 2}{2}

Cancel the common factor:

2

= 2

= 2 a 22 - 1

= 2 a 22 - 1

Simplify

2 (- \frac{7}{2}) : - 7

2 (- \frac{7}{2})

Apply rule:

a (- b) = - ab

2 (- \frac{7}{2}) = - 2 \cdot \frac{7}{2}

= - 2 \cdot \frac{7}{2}

Convert

2

to fraction

: \frac{2}{1}

2

Convert element to fraction:

2 = \frac{2}{1}

= \frac{2}{1}

= - \frac{2}{1} \cdot \frac{7}{2}

Cross-cancel common factor:

2

= - \frac{7}{1}

Apply the fraction rule:

\frac{a}{1} = a

= - 7

2 a 22 - 1 = - 7

2 a 22 - 1 = - 7

2 a 22 - 1 = - 7

Divide both sides by

2 22 - 1

2 a 22 - 1 = - 7

Divide both sides by

2 22 - 1

\frac{2 a 2 2 - 1}{2 2 2 - 1} = \frac{- 7}{2 2 2 - 1}

Simplify

a = - \frac{7}{2 2 2 - 1}

a = - \frac{7}{2 2 2 - 1}

For

2 ab = - \frac{7}{2}

, subsitute

b

with

- \frac{2 2 - 1}{2} : a = - \frac{7}{2 ( - 2 2 - 1 )}

For

2 ab = - \frac{7}{2}

, subsitute

b

with

- \frac{2 2 - 1}{2}

2 a (- \frac{2 2 - 1}{2}) = - \frac{7}{2}

Solve

2 a (- \frac{2 2 - 1}{2}) = - \frac{7}{2} : a = - \frac{7}{2 ( - 2 2 - 1 )}

2 a (- \frac{2 2 - 1}{2}) = - \frac{7}{2}

Divide both sides by

2 (- \frac{2 2 - 1}{2})

2 a (- \frac{2 2 - 1}{2}) = - \frac{7}{2}

Divide both sides by

2 (- \frac{2 2 - 1}{2})

\frac{2 a ( - \frac{2 2 - 1}{2} )}{2 ( - \frac{2 2 - 1}{2} )} = \frac{- \frac{7}{2}}{2 ( - \frac{2 2 - 1}{2} )}

Simplify

\frac{2 a ( - \frac{2 2 - 1}{2} )}{2 ( - \frac{2 2 - 1}{2} )} = \frac{- \frac{7}{2}}{2 ( - \frac{2 2 - 1}{2} )}

Simplify

\frac{2 a ( - \frac{2 2 - 1}{2} )}{2 ( - \frac{2 2 - 1}{2} )} : a

\frac{2 a ( - \frac{2 2 - 1}{2} )}{2 ( - \frac{2 2 - 1}{2} )}

Simplify

\frac{2 a ( - \frac{2 2 - 1}{2} )}{2 ( - \frac{2 2 - 1}{2} )} : \frac{- 2 a \frac{2 2 - 1}{2}}{- 2 \cdot \frac{2 2 - 1}{2}}

\frac{2 a ( - \frac{2 2 - 1}{2} )}{2 ( - \frac{2 2 - 1}{2} )}

Apply rule:

a (- b) = - ab

2 a (- \frac{2 2 - 1}{2}) = - 2 a \frac{2 2 - 1}{2}

= \frac{- 2 a \frac{2 2 - 1}{2}}{2 ( - \frac{2 2 - 1}{2} )}

Apply rule:

a (- b) = - ab

2 (- \frac{2 2 - 1}{2}) = - 2 \cdot \frac{2 2 - 1}{2}

= \frac{- 2 a \frac{2 2 - 1}{2}}{- 2 \cdot \frac{2 2 - 1}{2}}

= \frac{- 2 a \frac{2 2 - 1}{2}}{- 2 \cdot \frac{2 2 - 1}{2}}

Cancel the common factor:

- 2

= \frac{a \frac{2 2 - 1}{2}}{\frac{2 2 - 1}{2}}

Cancel the common factor:

\frac{2 2 - 1}{2}

= a

Simplify

\frac{- \frac{7}{2}}{2 ( - \frac{2 2 - 1}{2} )} : - \frac{7}{2 ( - 2 2 - 1 )}

\frac{- \frac{7}{2}}{2 ( - \frac{2 2 - 1}{2} )}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{7}{2}}{2 ( - \frac{2 2 - 1}{2} )}

Apply rule:

a (- b) = - ab

2 (- \frac{2 2 - 1}{2}) = - 2 \cdot \frac{2 2 - 1}{2}

= - \frac{\frac{7}{2}}{- 2 \cdot \frac{2 2 - 1}{2}}

- 2 \cdot \frac{2 2 - 1}{2} = - 22 - 1

- 2 \cdot \frac{2 2 - 1}{2}

Convert

2

to fraction

: \frac{2}{1}

2

Convert element to fraction:

2 = \frac{2}{1}

= \frac{2}{1}

= - \frac{2}{1} \cdot \frac{2 2 - 1}{2}

Cross-cancel common factor:

2

= - \frac{2 2 - 1}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{2 2 - 1}{1} = 22 - 1

= - 22 - 1

= - \frac{\frac{7}{2}}{- 2 2 - 1}

Apply the fraction rule:

\frac{\frac{a}{b}}{c} = \frac{a}{b \cdot c}

\frac{\frac{7}{2}}{- 2 2 - 1} = \frac{7}{2 ( - 2 2 - 1 )}

= - \frac{7}{2 ( - 2 2 - 1 )}

a = - \frac{7}{2 ( - 2 2 - 1 )}

a = - \frac{7}{2 ( - 2 2 - 1 )}

a = - \frac{7}{2 ( - 2 2 - 1 )}

Verify solutions by plugging them into the original equations

Check the solutions by plugging them into

a^{2} - b^{2} = \frac{1}{2}

Remove the ones that don't agree with the equation.

Check the solution

a = - \frac{7}{2 ( - 2 2 - 1 )}, b = - \frac{2 2 - 1}{2} :

True

a^{2} - b^{2} = \frac{1}{2}

Plug in

a = - \frac{7}{2 ( - 2 2 - 1 )}, b = - \frac{2 2 - 1}{2}

- \frac{7}{2 ( - 2 2 - 1 )}^{2} - (- \frac{2 2 - 1}{2})^{2} = \frac{1}{2}

Refine

\frac{1}{2} = \frac{1}{2}

True

Check the solution

a = - \frac{7}{2 2 2 - 1}, b = \frac{2 2 - 1}{2} :

True

a^{2} - b^{2} = \frac{1}{2}

Plug in

a = - \frac{7}{2 2 2 - 1}, b = \frac{2 2 - 1}{2}

(- \frac{7}{2 2 2 - 1})^{2} - (\frac{2 2 - 1}{2})^{2} = \frac{1}{2}

Refine

\frac{1}{2} = \frac{1}{2}

True

Check the solutions by plugging them into

2 ab = - \frac{7}{2}

Remove the ones that don't agree with the equation.

Check the solution

a = - \frac{7}{2 ( - 2 2 - 1 )}, b = - \frac{2 2 - 1}{2} :

True

2 ab = - \frac{7}{2}

Plug in

a = - \frac{7}{2 ( - 2 2 - 1 )}, b = - \frac{2 2 - 1}{2}

2 - \frac{7}{2 ( - 2 2 - 1 )} (- \frac{2 2 - 1}{2}) = - \frac{7}{2}

Refine

- \frac{7}{2} = - \frac{7}{2}

True

Check the solution

a = - \frac{7}{2 2 2 - 1}, b = \frac{2 2 - 1}{2} :

True

2 ab = - \frac{7}{2}

Plug in

a = - \frac{7}{2 2 2 - 1}, b = \frac{2 2 - 1}{2}

2 (- \frac{7}{2 2 2 - 1}) \frac{2 2 - 1}{2} = - \frac{7}{2}

Refine

- \frac{7}{2} = - \frac{7}{2}

True

Therefore, the final solutions for

a^{2} - b^{2} = \frac{1}{2}, 2 ab = - \frac{7}{2}

are

a = - \frac{7}{2 2 2 - 1}, a = - \frac{7}{2 ( - 2 2 - 1 )}, b = \frac{2 2 - 1}{2} b = - \frac{2 2 - 1}{2}

Substitute back

u = a + bi

u = - \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i, u = - \frac{7}{2 ( - 2 2 - 1 )} - \frac{2 2 - 1}{2} i

The solutions are

u = \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i, u = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}} - \frac{2 2 - 1}{2} i, u = - \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i, u = - \frac{7}{2 ( - 2 2 - 1 )} - \frac{2 2 - 1}{2} i

Substitute back

u = cos (x)

cos (x) = \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i, cos (x) = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}} - \frac{2 2 - 1}{2} i, cos (x) = - \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i, cos (x) = - \frac{7}{2 ( - 2 2 - 1 )} - \frac{2 2 - 1}{2} i

cos (x) = \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i, cos (x) = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}} - \frac{2 2 - 1}{2} i, cos (x) = - \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i, cos (x) = - \frac{7}{2 ( - 2 2 - 1 )} - \frac{2 2 - 1}{2} i

cos (x) = \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i :

No Solution

cos (x) = \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i

Simplify

\frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i : \frac{7 ( - 1 + 2 2 + 2 - 2 + 4 2 )}{14} + i \frac{- 1 + 2 2}{2}

\frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i

\frac{7}{2 2 2 - 1} = \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14}

\frac{7}{2 2 2 - 1}

Multiply by the conjugate

\frac{2 2 - 1}{2 2 - 1}

= \frac{7 2 2 - 1}{2 2 2 - 1 2 2 - 1}

2 22 - 1 22 - 1 = 42 - 2

2 22 - 1 22 - 1

Apply radical rule:

a a = a

22 - 1 22 - 1 = 22 - 1

= 2 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 22, c = 1

= 2 \cdot 22 - 2 \cdot 1

Simplify

2 \cdot 22 - 2 \cdot 1 : 42 - 2

2 \cdot 22 - 2 \cdot 1

Multiply the numbers:

2 \cdot 2 = 4

= 42 - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 42 - 2

= 42 - 2

= \frac{7 2 2 - 1}{4 2 - 2}

Multiply by the conjugate

\frac{4 2 + 2}{4 2 + 2}

= \frac{7 2 2 - 1 ( 4 2 + 2 )}{( 4 2 - 2 ) ( 4 2 + 2 )}

(42 - 2) (42 + 2) = 28

(42 - 2) (42 + 2)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 42, b = 2

= (42)^{2} - 2^{2}

Simplify

(42)^{2} - 2^{2} : 28

(42)^{2} - 2^{2}

(42)^{2} = 32

(42)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} (2)^{2}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 4^{2} \cdot 2

4^{2} = 16

= 16 \cdot 2

Multiply the numbers:

16 \cdot 2 = 32

= 32

2^{2} = 4

2^{2}

2^{2} = 4

= 4

= 32 - 4

Subtract the numbers:

32 - 4 = 28

= 28

= 28

= \frac{7 ( 4 2 + 2 ) 2 2 - 1}{28}

Factor

42 + 2 : 2 (22 + 1)

42 + 2

Rewrite as

= 2 \cdot 22 + 2 \cdot 1

Factor out common term

2

= 2 (22 + 1)

= \frac{7 \cdot 2 ( 2 2 + 1 ) 2 2 - 1}{28}

Cancel the common factor:

2

= \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14}

= \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} + i \frac{2 2 - 1}{2}

Rewrite

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} + \frac{2 2 - 1}{2} i

in standard complex form:

\frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14} + \frac{2 2 - 1}{2} i

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} + \frac{2 2 - 1}{2} i

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} = \frac{2 4 2 - 2 + 2 2 - 1}{2 7}

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{14}

Factor

14 : 2 \cdot 7

Factor

14 = 2 \cdot 7

= \frac{7 ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7}

Cancel

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7} : \frac{( 2 2 + 1 ) 2 2 - 1}{2 7}

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7}

Apply radical rule:

7 = 7^{\frac{1}{2}}

= \frac{7 ^{\frac{1}{2}} ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{7 ^{\frac{1}{2}}}{7 ^{1}} = \frac{1}{7 ^{1 - \frac{1}{2}}}

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7 ^{\frac{1}{2}}}

Apply radical rule:

7^{\frac{1}{2}} = 7

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 7}

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 7}

Expand

(22 + 1) 22 - 1 : 2 42 - 2 + 22 - 1

(22 + 1) 22 - 1

Apply the distributive law:

a (b + c) = ab + a c

a = 22 - 1, b = 22, c = 1

= 22 - 1 \cdot 22 + 22 - 1 \cdot 1

= 22 22 - 1 + 1 \cdot 22 - 1

Simplify

22 22 - 1 + 1 \cdot 22 - 1 : 2 42 - 2 + 22 - 1

22 22 - 1 + 1 \cdot 22 - 1

22 22 - 1 = 2 42 - 2

22 22 - 1

Apply radical rule:

a b = a \cdot b

2 22 - 1 = 2 (22 - 1)

= 2 2 (22 - 1)

Expand

2 (22 - 1) : 42 - 2

2 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 22, c = 1

= 2 \cdot 22 - 2 \cdot 1

Simplify

2 \cdot 22 - 2 \cdot 1 : 42 - 2

2 \cdot 22 - 2 \cdot 1

Multiply the numbers:

2 \cdot 2 = 4

= 42 - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 42 - 2

= 42 - 2

= 2 42 - 2

1 \cdot 22 - 1 = 22 - 1

1 \cdot 22 - 1

Multiply:

1 \cdot 22 - 1 = 22 - 1

= 22 - 1

= 2 42 - 2 + 22 - 1

= 2 42 - 2 + 22 - 1

= \frac{2 4 2 - 2 + 2 2 - 1}{2 7}

\frac{2 2 - 1}{2} i = \frac{i 2 2 - 1}{2}

\frac{2 2 - 1}{2} i

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 2 - 1 i}{2}

= \frac{2 4 2 - 2 + 2 2 - 1}{2 7} + \frac{i 2 2 - 1}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{2 4 2 - 2 + 2 2 - 1}{2 7} = \frac{2 4 2 - 2}{2 7} + \frac{2 2 - 1}{2 7}

= \frac{2 4 2 - 2}{2 7} + \frac{2 2 - 1}{2 7} + \frac{i 2 2 - 1}{2}

\frac{2 4 2 - 2}{2 7} = \frac{4 2 - 2}{7}

\frac{2 4 2 - 2}{2 7}

Divide the numbers:

\frac{2}{2} = 1

= \frac{4 2 - 2}{7}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{4 2 - 2}{7}

= \frac{4 2 - 2}{7} + \frac{2 2 - 1}{2 7} + \frac{i 2 2 - 1}{2}

Group the real part and the imaginary part of the complex number

= \frac{4 2 - 2}{7} + \frac{2 2 - 1}{2 7} + \frac{2 2 - 1}{2} i

\frac{4 2 - 2}{7} + \frac{2 2 - 1}{2 7} = \frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14}

\frac{4 2 - 2}{7} + \frac{2 2 - 1}{2 7}

Convert element to fraction:

\frac{4 2 - 2}{7} = \frac{\frac{4 \cdot 2 - 2}{7} 2 7}{2 7}

= \frac{\frac{4 2 - 2}{7} \cdot 2 7}{2 7} + \frac{2 2 - 1}{2 7}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{\frac{4 2 - 2}{7} \cdot 2 7 + 2 2 - 1}{2 7}

\frac{4 2 - 2}{7} \cdot 27 = 2 42 - 2

\frac{4 2 - 2}{7} \cdot 27

Apply radical rule:

a b = a \cdot b

7 \frac{4 2 - 2}{7} = 7 \cdot \frac{4 2 - 2}{7}

= 2 7 \cdot \frac{4 2 - 2}{7}

\frac{4 2 - 2}{7} \cdot 7 = 42 - 2

\frac{4 2 - 2}{7} \cdot 7

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 4 2 - 2 ) \cdot 7}{7}

Cancel the common factor:

7

= 42 - 2

= 2 42 - 2

= \frac{2 4 2 - 2 + 2 2 - 1}{2 7}

Rationalize

\frac{2 4 2 - 2 + 2 2 - 1}{2 7} : \frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14}

\frac{2 4 2 - 2 + 2 2 - 1}{2 7}

Multiply by the conjugate

\frac{7}{7}

= \frac{( 4 2 - 2 \cdot 2 + 2 2 - 1 ) 7}{2 7 7}

277 = 14

277

Apply radical rule:

a a = a

77 = 7

= 2 \cdot 7

Multiply the numbers:

2 \cdot 7 = 14

= 14

= \frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14}

= \frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14}

= \frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14} + \frac{2 2 - 1}{2} i

= \frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14} + \frac{2 2 - 1}{2} i

No Solution

cos (x) = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}} - \frac{2 2 - 1}{2} i :

No Solution

cos (x) = - \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}} - \frac{2 2 - 1}{2} i

Simplify

- \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}} - \frac{2 2 - 1}{2} i : \frac{7 ( - - 1 + 2 2 - 2 - 2 + 4 2 )}{14} - i \frac{- 1 + 2 2}{2}

- \frac{7}{2 ^{2} \cdot \frac{2 2 - 1}{2}} - \frac{2 2 - 1}{2} i

Multiply

2^{2} \cdot \frac{2 2 - 1}{2} : 2 22 - 1

2^{2} \cdot \frac{2 2 - 1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 2 - 1 \cdot 2 ^{2}}{2}

Cancel the common factor:

2

= 2 22 - 1

= - \frac{7}{2 2 2 - 1} - i \frac{2 2 - 1}{2}

\frac{7}{2 2 2 - 1} = \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14}

\frac{7}{2 2 2 - 1}

Multiply by the conjugate

\frac{2 2 - 1}{2 2 - 1}

= \frac{7 2 2 - 1}{2 2 2 - 1 2 2 - 1}

2 22 - 1 22 - 1 = 42 - 2

2 22 - 1 22 - 1

Apply radical rule:

a a = a

22 - 1 22 - 1 = 22 - 1

= 2 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 22, c = 1

= 2 \cdot 22 - 2 \cdot 1

Simplify

2 \cdot 22 - 2 \cdot 1 : 42 - 2

2 \cdot 22 - 2 \cdot 1

Multiply the numbers:

2 \cdot 2 = 4

= 42 - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 42 - 2

= 42 - 2

= \frac{7 2 2 - 1}{4 2 - 2}

Multiply by the conjugate

\frac{4 2 + 2}{4 2 + 2}

= \frac{7 2 2 - 1 ( 4 2 + 2 )}{( 4 2 - 2 ) ( 4 2 + 2 )}

(42 - 2) (42 + 2) = 28

(42 - 2) (42 + 2)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 42, b = 2

= (42)^{2} - 2^{2}

Simplify

(42)^{2} - 2^{2} : 28

(42)^{2} - 2^{2}

(42)^{2} = 32

(42)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} (2)^{2}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 4^{2} \cdot 2

4^{2} = 16

= 16 \cdot 2

Multiply the numbers:

16 \cdot 2 = 32

= 32

2^{2} = 4

2^{2}

2^{2} = 4

= 4

= 32 - 4

Subtract the numbers:

32 - 4 = 28

= 28

= 28

= \frac{7 ( 4 2 + 2 ) 2 2 - 1}{28}

Factor

42 + 2 : 2 (22 + 1)

42 + 2

Rewrite as

= 2 \cdot 22 + 2 \cdot 1

Factor out common term

2

= 2 (22 + 1)

= \frac{7 \cdot 2 ( 2 2 + 1 ) 2 2 - 1}{28}

Cancel the common factor:

2

= \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14}

= - \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} - i \frac{2 2 - 1}{2}

Rewrite

- \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} - \frac{2 2 - 1}{2} i

in standard complex form:

\frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14} - \frac{2 2 - 1}{2} i

- \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} - \frac{2 2 - 1}{2} i

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} = \frac{2 4 2 - 2 + 2 2 - 1}{2 7}

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{14}

Factor

14 : 2 \cdot 7

Factor

14 = 2 \cdot 7

= \frac{7 ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7}

Cancel

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7} : \frac{( 2 2 + 1 ) 2 2 - 1}{2 7}

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7}

Apply radical rule:

7 = 7^{\frac{1}{2}}

= \frac{7 ^{\frac{1}{2}} ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{7 ^{\frac{1}{2}}}{7 ^{1}} = \frac{1}{7 ^{1 - \frac{1}{2}}}

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7 ^{\frac{1}{2}}}

Apply radical rule:

7^{\frac{1}{2}} = 7

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 7}

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 7}

Expand

(22 + 1) 22 - 1 : 2 42 - 2 + 22 - 1

(22 + 1) 22 - 1

Apply the distributive law:

a (b + c) = ab + a c

a = 22 - 1, b = 22, c = 1

= 22 - 1 \cdot 22 + 22 - 1 \cdot 1

= 22 22 - 1 + 1 \cdot 22 - 1

Simplify

22 22 - 1 + 1 \cdot 22 - 1 : 2 42 - 2 + 22 - 1

22 22 - 1 + 1 \cdot 22 - 1

22 22 - 1 = 2 42 - 2

22 22 - 1

Apply radical rule:

a b = a \cdot b

2 22 - 1 = 2 (22 - 1)

= 2 2 (22 - 1)

Expand

2 (22 - 1) : 42 - 2

2 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 22, c = 1

= 2 \cdot 22 - 2 \cdot 1

Simplify

2 \cdot 22 - 2 \cdot 1 : 42 - 2

2 \cdot 22 - 2 \cdot 1

Multiply the numbers:

2 \cdot 2 = 4

= 42 - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 42 - 2

= 42 - 2

= 2 42 - 2

1 \cdot 22 - 1 = 22 - 1

1 \cdot 22 - 1

Multiply:

1 \cdot 22 - 1 = 22 - 1

= 22 - 1

= 2 42 - 2 + 22 - 1

= 2 42 - 2 + 22 - 1

= \frac{2 4 2 - 2 + 2 2 - 1}{2 7}

\frac{2 2 - 1}{2} i = \frac{i 2 2 - 1}{2}

\frac{2 2 - 1}{2} i

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 2 - 1 i}{2}

= - \frac{2 4 2 - 2 + 2 2 - 1}{2 7} - \frac{i 2 2 - 1}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{2 4 2 - 2 + 2 2 - 1}{2 7} = - (\frac{2 4 2 - 2}{2 7}) - (\frac{2 2 - 1}{2 7})

= - (\frac{2 4 2 - 2}{2 7}) - (\frac{2 2 - 1}{2 7}) - \frac{i 2 2 - 1}{2}

Remove parentheses:

(a) = a

= - \frac{2 4 2 - 2}{2 7} - \frac{2 2 - 1}{2 7} - \frac{i 2 2 - 1}{2}

\frac{2 4 2 - 2}{2 7} = \frac{4 2 - 2}{7}

\frac{2 4 2 - 2}{2 7}

Divide the numbers:

\frac{2}{2} = 1

= \frac{4 2 - 2}{7}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{4 2 - 2}{7}

= - \frac{4 2 - 2}{7} - \frac{2 2 - 1}{2 7} - \frac{i 2 2 - 1}{2}

Group the real part and the imaginary part of the complex number

= - \frac{4 2 - 2}{7} - \frac{2 2 - 1}{2 7} - \frac{2 2 - 1}{2} i

- \frac{4 2 - 2}{7} - \frac{2 2 - 1}{2 7} = \frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14}

- \frac{4 2 - 2}{7} - \frac{2 2 - 1}{2 7}

Convert element to fraction:

\frac{4 2 - 2}{7} = \frac{\frac{4 \cdot 2 - 2}{7} 2 7}{2 7}

= - \frac{\frac{4 2 - 2}{7} \cdot 2 7}{2 7} - \frac{2 2 - 1}{2 7}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- \frac{4 2 - 2}{7} \cdot 2 7 - 2 2 - 1}{2 7}

\frac{4 2 - 2}{7} \cdot 27 = 2 42 - 2

\frac{4 2 - 2}{7} \cdot 27

Apply radical rule:

a b = a \cdot b

7 \frac{4 2 - 2}{7} = 7 \cdot \frac{4 2 - 2}{7}

= 2 7 \cdot \frac{4 2 - 2}{7}

\frac{4 2 - 2}{7} \cdot 7 = 42 - 2

\frac{4 2 - 2}{7} \cdot 7

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 4 2 - 2 ) \cdot 7}{7}

Cancel the common factor:

7

= 42 - 2

= 2 42 - 2

= \frac{- 2 4 2 - 2 - 2 2 - 1}{2 7}

Rationalize

\frac{- 2 4 2 - 2 - 2 2 - 1}{2 7} : \frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14}

\frac{- 2 4 2 - 2 - 2 2 - 1}{2 7}

Multiply by the conjugate

\frac{7}{7}

= \frac{( - 4 2 - 2 \cdot 2 - 2 2 - 1 ) 7}{2 7 7}

277 = 14

277

Apply radical rule:

a a = a

77 = 7

= 2 \cdot 7

Multiply the numbers:

2 \cdot 7 = 14

= 14

= \frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14}

= \frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14}

= \frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14} - \frac{2 2 - 1}{2} i

= \frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14} - \frac{2 2 - 1}{2} i

No Solution

cos (x) = - \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i :

No Solution

cos (x) = - \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i

Simplify

- \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i : \frac{7 ( - - 1 + 2 2 - 2 - 2 + 4 2 )}{14} + i \frac{- 1 + 2 2}{2}

- \frac{7}{2 2 2 - 1} + \frac{2 2 - 1}{2} i

\frac{7}{2 2 2 - 1} = \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14}

\frac{7}{2 2 2 - 1}

Multiply by the conjugate

\frac{2 2 - 1}{2 2 - 1}

= \frac{7 2 2 - 1}{2 2 2 - 1 2 2 - 1}

2 22 - 1 22 - 1 = 42 - 2

2 22 - 1 22 - 1

Apply radical rule:

a a = a

22 - 1 22 - 1 = 22 - 1

= 2 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 22, c = 1

= 2 \cdot 22 - 2 \cdot 1

Simplify

2 \cdot 22 - 2 \cdot 1 : 42 - 2

2 \cdot 22 - 2 \cdot 1

Multiply the numbers:

2 \cdot 2 = 4

= 42 - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 42 - 2

= 42 - 2

= \frac{7 2 2 - 1}{4 2 - 2}

Multiply by the conjugate

\frac{4 2 + 2}{4 2 + 2}

= \frac{7 2 2 - 1 ( 4 2 + 2 )}{( 4 2 - 2 ) ( 4 2 + 2 )}

(42 - 2) (42 + 2) = 28

(42 - 2) (42 + 2)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 42, b = 2

= (42)^{2} - 2^{2}

Simplify

(42)^{2} - 2^{2} : 28

(42)^{2} - 2^{2}

(42)^{2} = 32

(42)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} (2)^{2}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 4^{2} \cdot 2

4^{2} = 16

= 16 \cdot 2

Multiply the numbers:

16 \cdot 2 = 32

= 32

2^{2} = 4

2^{2}

2^{2} = 4

= 4

= 32 - 4

Subtract the numbers:

32 - 4 = 28

= 28

= 28

= \frac{7 ( 4 2 + 2 ) 2 2 - 1}{28}

Factor

42 + 2 : 2 (22 + 1)

42 + 2

Rewrite as

= 2 \cdot 22 + 2 \cdot 1

Factor out common term

2

= 2 (22 + 1)

= \frac{7 \cdot 2 ( 2 2 + 1 ) 2 2 - 1}{28}

Cancel the common factor:

2

= \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14}

= - \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} + i \frac{2 2 - 1}{2}

Rewrite

- \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} + \frac{2 2 - 1}{2} i

in standard complex form:

\frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14} + \frac{2 2 - 1}{2} i

- \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} + \frac{2 2 - 1}{2} i

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} = \frac{2 4 2 - 2 + 2 2 - 1}{2 7}

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{14}

Factor

14 : 2 \cdot 7

Factor

14 = 2 \cdot 7

= \frac{7 ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7}

Cancel

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7} : \frac{( 2 2 + 1 ) 2 2 - 1}{2 7}

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7}

Apply radical rule:

7 = 7^{\frac{1}{2}}

= \frac{7 ^{\frac{1}{2}} ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{7 ^{\frac{1}{2}}}{7 ^{1}} = \frac{1}{7 ^{1 - \frac{1}{2}}}

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7 ^{\frac{1}{2}}}

Apply radical rule:

7^{\frac{1}{2}} = 7

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 7}

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 7}

Expand

(22 + 1) 22 - 1 : 2 42 - 2 + 22 - 1

(22 + 1) 22 - 1

Apply the distributive law:

a (b + c) = ab + a c

a = 22 - 1, b = 22, c = 1

= 22 - 1 \cdot 22 + 22 - 1 \cdot 1

= 22 22 - 1 + 1 \cdot 22 - 1

Simplify

22 22 - 1 + 1 \cdot 22 - 1 : 2 42 - 2 + 22 - 1

22 22 - 1 + 1 \cdot 22 - 1

22 22 - 1 = 2 42 - 2

22 22 - 1

Apply radical rule:

a b = a \cdot b

2 22 - 1 = 2 (22 - 1)

= 2 2 (22 - 1)

Expand

2 (22 - 1) : 42 - 2

2 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 22, c = 1

= 2 \cdot 22 - 2 \cdot 1

Simplify

2 \cdot 22 - 2 \cdot 1 : 42 - 2

2 \cdot 22 - 2 \cdot 1

Multiply the numbers:

2 \cdot 2 = 4

= 42 - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 42 - 2

= 42 - 2

= 2 42 - 2

1 \cdot 22 - 1 = 22 - 1

1 \cdot 22 - 1

Multiply:

1 \cdot 22 - 1 = 22 - 1

= 22 - 1

= 2 42 - 2 + 22 - 1

= 2 42 - 2 + 22 - 1

= \frac{2 4 2 - 2 + 2 2 - 1}{2 7}

\frac{2 2 - 1}{2} i = \frac{i 2 2 - 1}{2}

\frac{2 2 - 1}{2} i

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 2 - 1 i}{2}

= - \frac{2 4 2 - 2 + 2 2 - 1}{2 7} + \frac{i 2 2 - 1}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{2 4 2 - 2 + 2 2 - 1}{2 7} = - (\frac{2 4 2 - 2}{2 7}) - (\frac{2 2 - 1}{2 7})

= - (\frac{2 4 2 - 2}{2 7}) - (\frac{2 2 - 1}{2 7}) + \frac{i 2 2 - 1}{2}

Remove parentheses:

(a) = a

= - \frac{2 4 2 - 2}{2 7} - \frac{2 2 - 1}{2 7} + \frac{i 2 2 - 1}{2}

\frac{2 4 2 - 2}{2 7} = \frac{4 2 - 2}{7}

\frac{2 4 2 - 2}{2 7}

Divide the numbers:

\frac{2}{2} = 1

= \frac{4 2 - 2}{7}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{4 2 - 2}{7}

= - \frac{4 2 - 2}{7} - \frac{2 2 - 1}{2 7} + \frac{i 2 2 - 1}{2}

Group the real part and the imaginary part of the complex number

= - \frac{4 2 - 2}{7} - \frac{2 2 - 1}{2 7} + \frac{2 2 - 1}{2} i

- \frac{4 2 - 2}{7} - \frac{2 2 - 1}{2 7} = \frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14}

- \frac{4 2 - 2}{7} - \frac{2 2 - 1}{2 7}

Convert element to fraction:

\frac{4 2 - 2}{7} = \frac{\frac{4 \cdot 2 - 2}{7} 2 7}{2 7}

= - \frac{\frac{4 2 - 2}{7} \cdot 2 7}{2 7} - \frac{2 2 - 1}{2 7}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- \frac{4 2 - 2}{7} \cdot 2 7 - 2 2 - 1}{2 7}

\frac{4 2 - 2}{7} \cdot 27 = 2 42 - 2

\frac{4 2 - 2}{7} \cdot 27

Apply radical rule:

a b = a \cdot b

7 \frac{4 2 - 2}{7} = 7 \cdot \frac{4 2 - 2}{7}

= 2 7 \cdot \frac{4 2 - 2}{7}

\frac{4 2 - 2}{7} \cdot 7 = 42 - 2

\frac{4 2 - 2}{7} \cdot 7

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 4 2 - 2 ) \cdot 7}{7}

Cancel the common factor:

7

= 42 - 2

= 2 42 - 2

= \frac{- 2 4 2 - 2 - 2 2 - 1}{2 7}

Rationalize

\frac{- 2 4 2 - 2 - 2 2 - 1}{2 7} : \frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14}

\frac{- 2 4 2 - 2 - 2 2 - 1}{2 7}

Multiply by the conjugate

\frac{7}{7}

= \frac{( - 4 2 - 2 \cdot 2 - 2 2 - 1 ) 7}{2 7 7}

277 = 14

277

Apply radical rule:

a a = a

77 = 7

= 2 \cdot 7

Multiply the numbers:

2 \cdot 7 = 14

= 14

= \frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14}

= \frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14}

= \frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14} + \frac{2 2 - 1}{2} i

= \frac{7 ( - 2 4 2 - 2 - 2 2 - 1 )}{14} + \frac{2 2 - 1}{2} i

No Solution

cos (x) = - \frac{7}{2 ( - 2 2 - 1 )} - \frac{2 2 - 1}{2} i :

No Solution

cos (x) = - \frac{7}{2 ( - 2 2 - 1 )} - \frac{2 2 - 1}{2} i

Simplify

- \frac{7}{2 ( - 2 2 - 1 )} - \frac{2 2 - 1}{2} i : \frac{7 ( - 1 + 2 2 + 2 - 2 + 4 2 )}{14} - i \frac{- 1 + 2 2}{2}

- \frac{7}{2 ( - 2 2 - 1 )} - \frac{2 2 - 1}{2} i

Remove parentheses:

(- a) = - a

= - \frac{7}{- 2 2 2 - 1} - \frac{2 2 - 1}{2} i

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - (- \frac{7}{2 2 2 - 1}) - i \frac{2 2 - 1}{2}

Apply rule

- (- a) = a

= \frac{7}{2 2 2 - 1} - \frac{2 2 - 1}{2} i

\frac{7}{2 2 2 - 1} = \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14}

\frac{7}{2 2 2 - 1}

Multiply by the conjugate

\frac{2 2 - 1}{2 2 - 1}

= \frac{7 2 2 - 1}{2 2 2 - 1 2 2 - 1}

2 22 - 1 22 - 1 = 42 - 2

2 22 - 1 22 - 1

Apply radical rule:

a a = a

22 - 1 22 - 1 = 22 - 1

= 2 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 22, c = 1

= 2 \cdot 22 - 2 \cdot 1

Simplify

2 \cdot 22 - 2 \cdot 1 : 42 - 2

2 \cdot 22 - 2 \cdot 1

Multiply the numbers:

2 \cdot 2 = 4

= 42 - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 42 - 2

= 42 - 2

= \frac{7 2 2 - 1}{4 2 - 2}

Multiply by the conjugate

\frac{4 2 + 2}{4 2 + 2}

= \frac{7 2 2 - 1 ( 4 2 + 2 )}{( 4 2 - 2 ) ( 4 2 + 2 )}

(42 - 2) (42 + 2) = 28

(42 - 2) (42 + 2)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 42, b = 2

= (42)^{2} - 2^{2}

Simplify

(42)^{2} - 2^{2} : 28

(42)^{2} - 2^{2}

(42)^{2} = 32

(42)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} (2)^{2}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 4^{2} \cdot 2

4^{2} = 16

= 16 \cdot 2

Multiply the numbers:

16 \cdot 2 = 32

= 32

2^{2} = 4

2^{2}

2^{2} = 4

= 4

= 32 - 4

Subtract the numbers:

32 - 4 = 28

= 28

= 28

= \frac{7 ( 4 2 + 2 ) 2 2 - 1}{28}

Factor

42 + 2 : 2 (22 + 1)

42 + 2

Rewrite as

= 2 \cdot 22 + 2 \cdot 1

Factor out common term

2

= 2 (22 + 1)

= \frac{7 \cdot 2 ( 2 2 + 1 ) 2 2 - 1}{28}

Cancel the common factor:

2

= \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14}

= \frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} - i \frac{2 2 - 1}{2}

Rewrite

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} - \frac{2 2 - 1}{2} i

in standard complex form:

\frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14} - \frac{2 2 - 1}{2} i

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} - \frac{2 2 - 1}{2} i

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{14} = \frac{2 4 2 - 2 + 2 2 - 1}{2 7}

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{14}

Factor

14 : 2 \cdot 7

Factor

14 = 2 \cdot 7

= \frac{7 ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7}

Cancel

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7} : \frac{( 2 2 + 1 ) 2 2 - 1}{2 7}

\frac{7 ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7}

Apply radical rule:

7 = 7^{\frac{1}{2}}

= \frac{7 ^{\frac{1}{2}} ( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{7 ^{\frac{1}{2}}}{7 ^{1}} = \frac{1}{7 ^{1 - \frac{1}{2}}}

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 \cdot 7 ^{\frac{1}{2}}}

Apply radical rule:

7^{\frac{1}{2}} = 7

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 7}

= \frac{( 2 2 + 1 ) 2 2 - 1}{2 7}

Expand

(22 + 1) 22 - 1 : 2 42 - 2 + 22 - 1

(22 + 1) 22 - 1

Apply the distributive law:

a (b + c) = ab + a c

a = 22 - 1, b = 22, c = 1

= 22 - 1 \cdot 22 + 22 - 1 \cdot 1

= 22 22 - 1 + 1 \cdot 22 - 1

Simplify

22 22 - 1 + 1 \cdot 22 - 1 : 2 42 - 2 + 22 - 1

22 22 - 1 + 1 \cdot 22 - 1

22 22 - 1 = 2 42 - 2

22 22 - 1

Apply radical rule:

a b = a \cdot b

2 22 - 1 = 2 (22 - 1)

= 2 2 (22 - 1)

Expand

2 (22 - 1) : 42 - 2

2 (22 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 22, c = 1

= 2 \cdot 22 - 2 \cdot 1

Simplify

2 \cdot 22 - 2 \cdot 1 : 42 - 2

2 \cdot 22 - 2 \cdot 1

Multiply the numbers:

2 \cdot 2 = 4

= 42 - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 42 - 2

= 42 - 2

= 2 42 - 2

1 \cdot 22 - 1 = 22 - 1

1 \cdot 22 - 1

Multiply:

1 \cdot 22 - 1 = 22 - 1

= 22 - 1

= 2 42 - 2 + 22 - 1

= 2 42 - 2 + 22 - 1

= \frac{2 4 2 - 2 + 2 2 - 1}{2 7}

\frac{2 2 - 1}{2} i = \frac{i 2 2 - 1}{2}

\frac{2 2 - 1}{2} i

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 2 - 1 i}{2}

= \frac{2 4 2 - 2 + 2 2 - 1}{2 7} - \frac{i 2 2 - 1}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{2 4 2 - 2 + 2 2 - 1}{2 7} = \frac{2 4 2 - 2}{2 7} + \frac{2 2 - 1}{2 7}

= \frac{2 4 2 - 2}{2 7} + \frac{2 2 - 1}{2 7} - \frac{i 2 2 - 1}{2}

\frac{2 4 2 - 2}{2 7} = \frac{4 2 - 2}{7}

\frac{2 4 2 - 2}{2 7}

Divide the numbers:

\frac{2}{2} = 1

= \frac{4 2 - 2}{7}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{4 2 - 2}{7}

= \frac{4 2 - 2}{7} + \frac{2 2 - 1}{2 7} - \frac{i 2 2 - 1}{2}

Group the real part and the imaginary part of the complex number

= \frac{4 2 - 2}{7} + \frac{2 2 - 1}{2 7} - \frac{2 2 - 1}{2} i

\frac{4 2 - 2}{7} + \frac{2 2 - 1}{2 7} = \frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14}

\frac{4 2 - 2}{7} + \frac{2 2 - 1}{2 7}

Convert element to fraction:

\frac{4 2 - 2}{7} = \frac{\frac{4 \cdot 2 - 2}{7} 2 7}{2 7}

= \frac{\frac{4 2 - 2}{7} \cdot 2 7}{2 7} + \frac{2 2 - 1}{2 7}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{\frac{4 2 - 2}{7} \cdot 2 7 + 2 2 - 1}{2 7}

\frac{4 2 - 2}{7} \cdot 27 = 2 42 - 2

\frac{4 2 - 2}{7} \cdot 27

Apply radical rule:

a b = a \cdot b

7 \frac{4 2 - 2}{7} = 7 \cdot \frac{4 2 - 2}{7}

= 2 7 \cdot \frac{4 2 - 2}{7}

\frac{4 2 - 2}{7} \cdot 7 = 42 - 2

\frac{4 2 - 2}{7} \cdot 7

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 4 2 - 2 ) \cdot 7}{7}

Cancel the common factor:

7

= 42 - 2

= 2 42 - 2

= \frac{2 4 2 - 2 + 2 2 - 1}{2 7}

Rationalize

\frac{2 4 2 - 2 + 2 2 - 1}{2 7} : \frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14}

\frac{2 4 2 - 2 + 2 2 - 1}{2 7}

Multiply by the conjugate

\frac{7}{7}

= \frac{( 4 2 - 2 \cdot 2 + 2 2 - 1 ) 7}{2 7 7}

277 = 14

277

Apply radical rule:

a a = a

77 = 7

= 2 \cdot 7

Multiply the numbers:

2 \cdot 7 = 14

= 14

= \frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14}

= \frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14}

= \frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14} - \frac{2 2 - 1}{2} i

= \frac{7 ( 2 4 2 - 2 + 2 2 - 1 )}{14} - \frac{2 2 - 1}{2} i

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for 1+sin^2(x)+cos^4(x)=0 ?
The general solution for 1+sin^2(x)+cos^4(x)=0 is No Solution for x\in\mathbb{R}