What is the general solution for 2sech^2(x)+tanh(x)=0 ?

The general solution for 2sech^2(x)+tanh(x)=0 is x= 1/2 ln(-4+sqrt(17))

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2sech^2(x)+tanh(x)=0

Solution

2 sech^{2} (x) + tanh (x) = 0

Solution

x = \frac{1}{2} ln (- 4 + 17)

Degrees

x = - 60.00909 \dots^{\circ}

Solution steps

2 sech^{2} (x) + tanh (x) = 0

Rewrite using trig identities

2 sech^{2} (x) + tanh (x) = 0

Use the Hyperbolic identity:

tanh (x) = \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}

2 sech^{2} (x) + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0

Use the Hyperbolic identity:

sech (x) = \frac{2}{e ^{x} + e ^{- x}}

2 (\frac{2}{e ^{x} + e ^{- x}})^{2} + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0

2 (\frac{2}{e ^{x} + e ^{- x}})^{2} + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0

2 (\frac{2}{e ^{x} + e ^{- x}})^{2} + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0 : x = \frac{1}{2} ln (- 4 + 17)

2 (\frac{2}{e ^{x} + e ^{- x}})^{2} + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0

Multiply both sides by

e^{x} + e^{- x}

2 (\frac{2}{e ^{x} + e ^{- x}})^{2} (e^{x} + e^{- x}) + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} (e^{x} + e^{- x}) = 0 \cdot (e^{x} + e^{- x})

Simplify

2 (\frac{2}{e ^{x} + e ^{- x}})^{2} (e^{x} + e^{- x}) + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} (e^{x} + e^{- x}) : \frac{8}{e ^{x} + e ^{- x}} + e^{x} - e^{- x}

2 (\frac{2}{e ^{x} + e ^{- x}})^{2} (e^{x} + e^{- x}) + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} (e^{x} + e^{- x})

2 (\frac{2}{e ^{x} + e ^{- x}})^{2} (e^{x} + e^{- x}) = \frac{8}{e ^{x} + e ^{- x}}

2 (\frac{2}{e ^{x} + e ^{- x}})^{2} (e^{x} + e^{- x})

(\frac{2}{e ^{x} + e ^{- x}})^{2} = \frac{2 ^{2}}{( e ^{x} + e ^{- x} ) ^{2}}

(\frac{2}{e ^{x} + e ^{- x}})^{2}

Apply exponent rule:

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

= \frac{2 ^{2}}{( e ^{x} + e ^{- x} ) ^{2}}

= 2 \cdot \frac{2 ^{2}}{( e ^{x} + e ^{- x} ) ^{2}} (e^{x} + e^{- x})

Multiply fractions:

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

= \frac{2 ^{2} \cdot 2 ( e ^{x} + e ^{- x} )}{( e ^{x} + e ^{- x} ) ^{2}}

2^{2} \cdot 2 (e^{x} + e^{- x}) = 2^{3} (e^{x} + e^{- x})

2^{2} \cdot 2 (e^{x} + e^{- x})

Apply exponent rule:

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

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

= 2^{2 + 1} (e^{x} + e^{- x})

Add the numbers:

2 + 1 = 3

= 2^{3} (e^{x} + e^{- x})

= \frac{2 ^{3} ( e ^{x} + e ^{- x} )}{( e ^{x} + e ^{- x} ) ^{2}}

Cancel the common factor:

e^{x} + e^{- x}

= \frac{2 ^{3}}{e ^{x} + e ^{- x}}

2^{3} = 8

= \frac{8}{e ^{x} + e ^{- x}}

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} (e^{x} + e^{- x}) = e^{x} - e^{- x}

\frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} (e^{x} + e^{- x})

Multiply fractions:

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

= \frac{( e ^{x} - e ^{- x} ) ( e ^{x} + e ^{- x} )}{e ^{x} + e ^{- x}}

Cancel the common factor:

e^{x} + e^{- x}

= e^{x} - e^{- x}

= \frac{8}{e ^{x} + e ^{- x}} + e^{x} - e^{- x}

\frac{8}{e ^{x} + e ^{- x}} + e^{x} - e^{- x} = 0

Apply exponent rules

\frac{8}{e ^{x} + e ^{- x}} + e^{x} - e^{- x} = 0

Apply exponent rule:

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

e^{- x} = (e^{x})^{- 1}

\frac{8}{e ^{x} + ( e ^{x} ) ^{- 1}} + e^{x} - (e^{x})^{- 1} = 0

\frac{8}{e ^{x} + ( e ^{x} ) ^{- 1}} + e^{x} - (e^{x})^{- 1} = 0

Rewrite the equation with

e^{x} = u

\frac{8}{u + ( u ) ^{- 1}} + u - (u)^{- 1} = 0

Solve

\frac{8}{u + u ^{- 1}} + u - u^{- 1} = 0 : u = - 4 + 17, u = - - 4 + 17

\frac{8}{u + u ^{- 1}} + u - u^{- 1} = 0

Refine

\frac{8 u}{u ^{2} + 1} + u - \frac{1}{u} = 0

Multiply by LCM

\frac{8 u}{u ^{2} + 1} + u - \frac{1}{u} = 0

Find Least Common Multiplier of

u^{2} + 1, u : u (u^{2} + 1)

u^{2} + 1, u

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

u^{2} + 1

u

= u (u^{2} + 1)

Multiply by LCM=

u (u^{2} + 1)

\frac{8 u}{u ^{2} + 1} u (u^{2} + 1) + uu (u^{2} + 1) - \frac{1}{u} u (u^{2} + 1) = 0 \cdot u (u^{2} + 1)

Simplify

\frac{8 u}{u ^{2} + 1} u (u^{2} + 1) + uu (u^{2} + 1) - \frac{1}{u} u (u^{2} + 1) = 0 \cdot u (u^{2} + 1)

Simplify

\frac{8 u}{u ^{2} + 1} u (u^{2} + 1) : 8 u^{2}

\frac{8 u}{u ^{2} + 1} u (u^{2} + 1)

Multiply fractions:

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

= \frac{8 uu ( u ^{2} + 1 )}{u ^{2} + 1}

Cancel the common factor:

u^{2} + 1

= 8 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 8 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 8 u^{2}

Simplify

uu (u^{2} + 1) : u^{2} (u^{2} + 1)

uu (u^{2} + 1)

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1} (u^{2} + 1)

Add the numbers:

1 + 1 = 2

= u^{2} (u^{2} + 1)

Simplify

- \frac{1}{u} u (u^{2} + 1) : - (u^{2} + 1)

- \frac{1}{u} u (u^{2} + 1)

Multiply fractions:

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

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

Cancel the common factor:

u

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

Multiply:

1 \cdot (u^{2} + 1) = (u^{2} + 1)

= - (u^{2} + 1)

Simplify

0 \cdot u (u^{2} + 1) : 0

0 \cdot u (u^{2} + 1)

Apply rule

0 \cdot a = 0

= 0

8 u^{2} + u^{2} (u^{2} + 1) - (u^{2} + 1) = 0

8 u^{2} + u^{2} (u^{2} + 1) - (u^{2} + 1) = 0

8 u^{2} + u^{2} (u^{2} + 1) - (u^{2} + 1) = 0

Solve

8 u^{2} + u^{2} (u^{2} + 1) - (u^{2} + 1) = 0 : u = - 4 + 17, u = - - 4 + 17

8 u^{2} + u^{2} (u^{2} + 1) - (u^{2} + 1) = 0

Expand

8 u^{2} + u^{2} (u^{2} + 1) - (u^{2} + 1) : u^{4} + 8 u^{2} - 1

8 u^{2} + u^{2} (u^{2} + 1) - (u^{2} + 1)

Expand

u^{2} (u^{2} + 1) : u^{4} + u^{2}

u^{2} (u^{2} + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = u^{2}, b = u^{2}, c = 1

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

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

Simplify

u^{2} u^{2} + 1 \cdot u^{2} : u^{4} + u^{2}

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

u^{2} u^{2} = u^{4}

u^{2} u^{2}

Apply exponent rule:

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

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

Add the numbers:

2 + 2 = 4

= u^{4}

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

1 \cdot u^{2}

Multiply:

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

= u^{2}

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

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

= 8 u^{2} + u^{4} + u^{2} - (u^{2} + 1)

- (u^{2} + 1) : - u^{2} - 1

- (u^{2} + 1)

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

= - u^{2} - 1

= 8 u^{2} + u^{4} + u^{2} - u^{2} - 1

Simplify

8 u^{2} + u^{4} + u^{2} - u^{2} - 1 : u^{4} + 8 u^{2} - 1

8 u^{2} + u^{4} + u^{2} - u^{2} - 1

Group like terms

= u^{4} + 8 u^{2} + u^{2} - u^{2} - 1

Add similar elements:

8 u^{2} + u^{2} - u^{2} = 8 u^{2}

= u^{4} + 8 u^{2} - 1

= u^{4} + 8 u^{2} - 1

u^{4} + 8 u^{2} - 1 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

v^{2} + 8 v - 1 = 0

Solve

v^{2} + 8 v - 1 = 0 : v = - 4 + 17, v = - 4 - 17

v^{2} + 8 v - 1 = 0

Solve with the quadratic formula

v^{2} + 8 v - 1 = 0

Quadratic Equation Formula:

For

a = 1, b = 8, c = - 1

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

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

8^{2} - 4 \cdot 1 \cdot (- 1) = 217

8^{2} - 4 \cdot 1 \cdot (- 1)

Apply rule

- (- a) = a

= 8^{2} + 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 8^{2} + 4

8^{2} = 64

= 64 + 4

Add the numbers:

64 + 4 = 68

= 68

Prime factorization of

68 : 2^{2} \cdot 17

68

68

divides by

268 = 34 \cdot 2

= 2 \cdot 34

34

divides by

234 = 17 \cdot 2

= 2 \cdot 2 \cdot 17

2, 17

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 17

= 2^{2} \cdot 17

= 2^{2} \cdot 17

Apply radical rule:

= 17 2^{2}

Apply radical rule:

2^{2} = 2

= 217

v_{1, 2} = \frac{- 8 \pm 2 17}{2 \cdot 1}

Separate the solutions

v_{1} = \frac{- 8 + 2 17}{2 \cdot 1}, v_{2} = \frac{- 8 - 2 17}{2 \cdot 1}

v = \frac{- 8 + 2 17}{2 \cdot 1} : - 4 + 17

\frac{- 8 + 2 17}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 8 + 2 17}{2}

Factor

- 8 + 217 : 2 (- 4 + 17)

- 8 + 217

Rewrite as

= - 2 \cdot 4 + 217

Factor out common term

2

= 2 (- 4 + 17)

= \frac{2 ( - 4 + 17 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 4 + 17

v = \frac{- 8 - 2 17}{2 \cdot 1} : - 4 - 17

\frac{- 8 - 2 17}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 8 - 2 17}{2}

Factor

- 8 - 217 : - 2 (4 + 17)

- 8 - 217

Rewrite as

= - 2 \cdot 4 - 217

Factor out common term

2

= - 2 (4 + 17)

= - \frac{2 ( 4 + 17 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - (4 + 17)

Negate

- (4 + 17) = - 4 - 17

= - 4 - 17

The solutions to the quadratic equation are:

v = - 4 + 17, v = - 4 - 17

v = - 4 + 17, v = - 4 - 17

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = - 4 + 17 : u = - 4 + 17, u = - - 4 + 17

u^{2} = - 4 + 17

For

x^{2} = f (a)

the solutions are

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

u = - 4 + 17, u = - - 4 + 17

Solve

u^{2} = - 4 - 17 :

No Solution for

u \in R

u^{2} = - 4 - 17

x^{2}

cannot be negative for

x \in R

No Solution for u \in R

The solutions are

u = - 4 + 17, u = - - 4 + 17

u = - 4 + 17, u = - - 4 + 17

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{8}{u + u ^{- 1}} + u - u^{- 1}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - 4 + 17, u = - - 4 + 17

u = - 4 + 17, u = - - 4 + 17

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = - 4 + 17 : x = \frac{1}{2} ln (- 4 + 17)

e^{x} = - 4 + 17

Apply exponent rules

e^{x} = - 4 + 17

Apply exponent rule:

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

- 4 + 17 = (- 4 + 17)^{\frac{1}{2}}

e^{x} = (- 4 + 17)^{\frac{1}{2}}

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln ((- 4 + 17)^{\frac{1}{2}})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln ((- 4 + 17)^{\frac{1}{2}})

Apply log rule:

ln (x^{a}) = a \cdot ln (x)

ln ((- 4 + 17)^{\frac{1}{2}}) = \frac{1}{2} ln (- 4 + 17)

x = \frac{1}{2} ln (- 4 + 17)

x = \frac{1}{2} ln (- 4 + 17)

Solve

e^{x} = - - 4 + 17 :

No Solution for

x \in R

e^{x} = - - 4 + 17

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = \frac{1}{2} ln (- 4 + 17)

Verify Solutions:

x = \frac{1}{2} ln (- 4 + 17)

True

Check the solutions by plugging them into

2 (\frac{2}{e ^{x} + e ^{- x}})^{2} + \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = 0

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

Plug in

x = \frac{1}{2} ln (- 4 + 17) :

True

2 (\frac{2}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}})^{2} + \frac{e ^{\frac{1}{2} l n (- 4 + 17)} - e ^{- \frac{1}{2} l n (- 4 + 17)}}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}} = 0

2 (\frac{2}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}})^{2} + \frac{e ^{\frac{1}{2} l n (- 4 + 17)} - e ^{- \frac{1}{2} l n (- 4 + 17)}}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}} = 0

2 (\frac{2}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}})^{2} + \frac{e ^{\frac{1}{2} l n (- 4 + 17)} - e ^{- \frac{1}{2} l n (- 4 + 17)}}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}}

2 (\frac{2}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}})^{2} = \frac{4 ( 17 - 4 )}{13 - 3 17}

2 (\frac{2}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}})^{2}

(\frac{2}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}})^{2} = \frac{2 ( 17 - 4 )}{13 - 3 17}

(\frac{2}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}})^{2}

\frac{2}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}} = \frac{2 - 4 + 17}{17 - 3}

\frac{2}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}}

e^{\frac{1}{2} l n (- 4 + 17)} = - 4 + 17

e^{\frac{1}{2} l n (- 4 + 17)}

Apply exponent rule:

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

= e^{l n (- 4 + 17)}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (- 4 + 17)} = - 4 + 17

= - 4 + 17

e^{- \frac{1}{2} l n (- 4 + 17)} = \frac{1}{- 4 + 17}

e^{- \frac{1}{2} l n (- 4 + 17)}

Apply exponent rule:

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

= (e^{l n (- 4 + 17)})^{- \frac{1}{2}}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (- 4 + 17)} = - 4 + 17

= (- 4 + 17)^{- \frac{1}{2}}

Apply exponent rule:

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

= \frac{1}{- 4 + 17}

= \frac{2}{17 - 4 + \frac{1}{17 - 4}}

Join

- 4 + 17 + \frac{1}{- 4 + 17} : \frac{17 - 3}{- 4 + 17}

- 4 + 17 + \frac{1}{- 4 + 17}

Convert element to fraction:

17 - 4 = \frac{- 4 + 17 - 4 + 17}{- 4 + 17}

= \frac{- 4 + 17 - 4 + 17}{- 4 + 17} + \frac{1}{- 4 + 17}

Since the denominators are equal, combine the fractions:

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

= \frac{- 4 + 17 - 4 + 17 + 1}{- 4 + 17}

- 4 + 17 - 4 + 17 + 1 = 17 - 3

- 4 + 17 - 4 + 17 + 1

Apply radical rule:

a a = a

17 - 4 17 - 4 = - 4 + 17

= (17 - 4) + 1

Remove parentheses:

(- a) = - a

= - 4 + 17 + 1

Add/Subtract the numbers:

- 4 + 1 = - 3

= 17 - 3

= \frac{17 - 3}{- 4 + 17}

= \frac{2}{\frac{17 - 3}{- 4 + 17}}

Apply the fraction rule:

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

= \frac{2 - 4 + 17}{17 - 3}

= (\frac{2 - 4 + 17}{17 - 3})^{2}

Apply exponent rule:

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

= \frac{( 2 17 - 4 ) ^{2}}{( 17 - 3 ) ^{2}}

Apply exponent rule:

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

(2 17 - 4)^{2} = 2^{2} (17 - 4)^{2}

= \frac{2 ^{2} ( 17 - 4 ) ^{2}}{( 17 - 3 ) ^{2}}

(- 4 + 17)^{2} : - 4 + 17

Apply radical rule:

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

= ((- 4 + 17)^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (- 4 + 17)^{\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

= - 4 + 17

= \frac{2 ^{2} ( - 4 + 17 )}{( 17 - 3 ) ^{2}}

(17 - 3)^{2} = 26 - 617

(17 - 3)^{2}

Apply Perfect Square Formula:

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

a = 17, b = 3

= (17)^{2} - 217 \cdot 3 + 3^{2}

Simplify

(17)^{2} - 217 \cdot 3 + 3^{2} : 26 - 617

(17)^{2} - 217 \cdot 3 + 3^{2}

(17)^{2} = 17

(17)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= 1 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

= 17

217 \cdot 3 = 617

217 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 617

3^{2} = 9

3^{2}

3^{2} = 9

= 9

= 17 - 617 + 9

Add the numbers:

17 + 9 = 26

= 26 - 617

= 26 - 617

= \frac{2 ^{2} ( 17 - 4 )}{26 - 6 17}

Factor

26 - 617 : 2 (13 - 317)

26 - 617

Rewrite as

= 2 \cdot 13 - 2 \cdot 317

Factor out common term

2

= 2 (13 - 317)

= \frac{2 ^{2} ( - 4 + 17 )}{2 ( 13 - 3 17 )}

Cancel the common factor:

2

= \frac{2 ( 17 - 4 )}{13 - 3 17}

= 2 \cdot \frac{2 ( 17 - 4 )}{13 - 3 17}

Multiply fractions:

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

= \frac{2 ( 17 - 4 ) \cdot 2}{13 - 3 17}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 ( 17 - 4 )}{13 - 3 17}

\frac{e ^{\frac{1}{2} l n (- 4 + 17)} - e ^{- \frac{1}{2} l n (- 4 + 17)}}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}} = \frac{17 - 5}{17 - 3}

\frac{e ^{\frac{1}{2} l n (- 4 + 17)} - e ^{- \frac{1}{2} l n (- 4 + 17)}}{e ^{\frac{1}{2} l n (- 4 + 17)} + e ^{- \frac{1}{2} l n (- 4 + 17)}}

e^{\frac{1}{2} l n (- 4 + 17)} = - 4 + 17

e^{\frac{1}{2} l n (- 4 + 17)}

Apply exponent rule:

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

= e^{l n (- 4 + 17)}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (- 4 + 17)} = - 4 + 17

= - 4 + 17

e^{- \frac{1}{2} l n (- 4 + 17)} = \frac{1}{- 4 + 17}

e^{- \frac{1}{2} l n (- 4 + 17)}

Apply exponent rule:

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

= (e^{l n (- 4 + 17)})^{- \frac{1}{2}}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (- 4 + 17)} = - 4 + 17

= (- 4 + 17)^{- \frac{1}{2}}

Apply exponent rule:

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

= \frac{1}{- 4 + 17}

= \frac{e ^{\frac{1}{2} l n (17 - 4)} - e ^{- \frac{1}{2} l n (17 - 4)}}{17 - 4 + \frac{1}{17 - 4}}

e^{\frac{1}{2} l n (- 4 + 17)} = - 4 + 17

e^{\frac{1}{2} l n (- 4 + 17)}

Apply exponent rule:

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

= e^{l n (- 4 + 17)}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (- 4 + 17)} = - 4 + 17

= - 4 + 17

e^{- \frac{1}{2} l n (- 4 + 17)} = \frac{1}{- 4 + 17}

e^{- \frac{1}{2} l n (- 4 + 17)}

Apply exponent rule:

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

= (e^{l n (- 4 + 17)})^{- \frac{1}{2}}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (- 4 + 17)} = - 4 + 17

= (- 4 + 17)^{- \frac{1}{2}}

Apply exponent rule:

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

= \frac{1}{- 4 + 17}

= \frac{17 - 4 - \frac{1}{17 - 4}}{17 - 4 + \frac{1}{17 - 4}}

Join

- 4 + 17 + \frac{1}{- 4 + 17} : \frac{17 - 3}{- 4 + 17}

- 4 + 17 + \frac{1}{- 4 + 17}

Convert element to fraction:

17 - 4 = \frac{- 4 + 17 - 4 + 17}{- 4 + 17}

= \frac{- 4 + 17 - 4 + 17}{- 4 + 17} + \frac{1}{- 4 + 17}

Since the denominators are equal, combine the fractions:

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

= \frac{- 4 + 17 - 4 + 17 + 1}{- 4 + 17}

- 4 + 17 - 4 + 17 + 1 = 17 - 3

- 4 + 17 - 4 + 17 + 1

Apply radical rule:

a a = a

17 - 4 17 - 4 = - 4 + 17

= (17 - 4) + 1

Remove parentheses:

(- a) = - a

= - 4 + 17 + 1

Add/Subtract the numbers:

- 4 + 1 = - 3

= 17 - 3

= \frac{17 - 3}{- 4 + 17}

= \frac{17 - 4 - \frac{1}{17 - 4}}{\frac{17 - 3}{- 4 + 17}}

Join

- 4 + 17 - \frac{1}{- 4 + 17} : \frac{17 - 5}{- 4 + 17}

- 4 + 17 - \frac{1}{- 4 + 17}

Convert element to fraction:

17 - 4 = \frac{- 4 + 17 - 4 + 17}{- 4 + 17}

= \frac{- 4 + 17 - 4 + 17}{- 4 + 17} - \frac{1}{- 4 + 17}

Since the denominators are equal, combine the fractions:

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

= \frac{- 4 + 17 - 4 + 17 - 1}{- 4 + 17}

- 4 + 17 - 4 + 17 - 1 = 17 - 5

- 4 + 17 - 4 + 17 - 1

Apply radical rule:

a a = a

17 - 4 17 - 4 = - 4 + 17

= (17 - 4) - 1

Remove parentheses:

(- a) = - a

= - 4 + 17 - 1

Subtract the numbers:

- 4 - 1 = - 5

= 17 - 5

= \frac{17 - 5}{- 4 + 17}

= \frac{\frac{17 - 5}{- 4 + 17}}{\frac{17 - 3}{- 4 + 17}}

Divide fractions:

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

= \frac{( 17 - 5 ) - 4 + 17}{- 4 + 17 ( 17 - 3 )}

Cancel the common factor:

- 4 + 17

= \frac{17 - 5}{17 - 3}

= \frac{4 ( 17 - 4 )}{13 - 3 17} + \frac{17 - 5}{17 - 3}

Simplify

\frac{4 ( 17 - 4 )}{13 - 3 17} + \frac{17 - 5}{17 - 3}

Least Common Multiplier of

13 - 317, 17 - 3 : (17 - 3) (13 - 317)

13 - 317, 17 - 3

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

13 - 317

17 - 3

= (17 - 3) (13 - 317)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

(17 - 3) (13 - 317)

For

\frac{4 ( 17 - 4 )}{13 - 3 17} :

multiply the denominator and numerator by

17 - 3

\frac{4 ( 17 - 4 )}{13 - 3 17} = \frac{4 ( 17 - 4 ) ( 17 - 3 )}{( 13 - 3 17 ) ( 17 - 3 )}

For

\frac{17 - 5}{17 - 3} :

multiply the denominator and numerator by

13 - 317

\frac{17 - 5}{17 - 3} = \frac{( 17 - 5 ) ( 13 - 3 17 )}{( 17 - 3 ) ( 13 - 3 17 )}

= \frac{4 ( 17 - 4 ) ( 17 - 3 )}{( 13 - 3 17 ) ( 17 - 3 )} + \frac{( 17 - 5 ) ( 13 - 3 17 )}{( 17 - 3 ) ( 13 - 3 17 )}

Since the denominators are equal, combine the fractions:

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

= \frac{4 ( 17 - 4 ) ( 17 - 3 ) + ( 17 - 5 ) ( 13 - 3 17 )}{( 17 - 3 ) ( 13 - 3 17 )}

Expand

(17 - 3) (13 - 317) : 2217 - 90

(17 - 3) (13 - 317)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 17, b = - 3, c = 13, d = - 317

= 17 \cdot 13 + 17 (- 317) + (- 3) \cdot 13 + (- 3) (- 317)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= 1317 - 31717 - 3 \cdot 13 + 3 \cdot 317

Simplify

1317 - 31717 - 3 \cdot 13 + 3 \cdot 317 : 2217 - 90

1317 - 31717 - 3 \cdot 13 + 3 \cdot 317

31717 = 51

31717

Apply radical rule:

a a = a

1717 = 17

= 3 \cdot 17

Multiply the numbers:

3 \cdot 17 = 51

= 51

3 \cdot 13 = 39

3 \cdot 13

Multiply the numbers:

3 \cdot 13 = 39

= 39

3 \cdot 317 = 917

3 \cdot 317

Multiply the numbers:

3 \cdot 3 = 9

= 917

= 1317 - 51 - 39 + 917

Add similar elements:

1317 + 917 = 2217

= 2217 - 51 - 39

Subtract the numbers:

- 51 - 39 = - 90

= 2217 - 90

= 2217 - 90

= \frac{4 ( 17 - 4 ) ( 17 - 3 ) + ( 17 - 5 ) ( 13 - 3 17 )}{22 17 - 90}

Expand

4 (17 - 4) (17 - 3) + (17 - 5) (13 - 317) : 0

4 (17 - 4) (17 - 3) + (17 - 5) (13 - 317)

Expand

4 (17 - 4) (17 - 3) : 116 - 2817

Expand

(17 - 4) (17 - 3) : 29 - 717

(17 - 4) (17 - 3)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 17, b = - 4, c = 17, d = - 3

= 1717 + 17 (- 3) + (- 4) 17 + (- 4) (- 3)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= 1717 - 317 - 417 + 4 \cdot 3

Simplify

1717 - 317 - 417 + 4 \cdot 3 : 29 - 717

1717 - 317 - 417 + 4 \cdot 3

Add similar elements:

- 317 - 417 = - 717

= 1717 - 717 + 4 \cdot 3

Apply radical rule:

a a = a

1717 = 17

= 17 - 717 + 4 \cdot 3

Multiply the numbers:

4 \cdot 3 = 12

= 17 - 717 + 12

Add the numbers:

17 + 12 = 29

= 29 - 717

= 29 - 717

= 4 (29 - 717)

Expand

4 (29 - 717) : 116 - 2817

4 (29 - 717)

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 29, c = 717

= 4 \cdot 29 - 4 \cdot 717

Simplify

4 \cdot 29 - 4 \cdot 717 : 116 - 2817

4 \cdot 29 - 4 \cdot 717

Multiply the numbers:

4 \cdot 29 = 116

= 116 - 4 \cdot 717

Multiply the numbers:

4 \cdot 7 = 28

= 116 - 2817

= 116 - 2817

= 116 - 2817

= 116 - 2817 + (17 - 5) (13 - 317)

Expand

(17 - 5) (13 - 317) : 2817 - 116

(17 - 5) (13 - 317)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 17, b = - 5, c = 13, d = - 317

= 17 \cdot 13 + 17 (- 317) + (- 5) \cdot 13 + (- 5) (- 317)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= 1317 - 31717 - 5 \cdot 13 + 5 \cdot 317

Simplify

1317 - 31717 - 5 \cdot 13 + 5 \cdot 317 : 2817 - 116

1317 - 31717 - 5 \cdot 13 + 5 \cdot 317

31717 = 51

31717

Apply radical rule:

a a = a

1717 = 17

= 3 \cdot 17

Multiply the numbers:

3 \cdot 17 = 51

= 51

5 \cdot 13 = 65

5 \cdot 13

Multiply the numbers:

5 \cdot 13 = 65

= 65

5 \cdot 317 = 1517

5 \cdot 317

Multiply the numbers:

5 \cdot 3 = 15

= 1517

= 1317 - 51 - 65 + 1517

Add similar elements:

1317 + 1517 = 2817

= 2817 - 51 - 65

Subtract the numbers:

- 51 - 65 = - 116

= 2817 - 116

= 2817 - 116

= 116 - 2817 + 2817 - 116

Simplify

116 - 2817 + 2817 - 116 : 0

116 - 2817 + 2817 - 116

Add similar elements:

- 2817 + 2817 = 0

= 116 - 116

Subtract the numbers:

116 - 116 = 0

= 0

= 0

= \frac{0}{22 17 - 90}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

= 0

0 = 0

True

The solution is

x = \frac{1}{2} ln (- 4 + 17)

x = \frac{1}{2} ln (- 4 + 17)

Popular Examples

8cos(2θ)=-24cos(θ)-16

8 cos (2 θ) = - 24 cos (θ) - 16

16cos^2(θ)-9=0

16 cos^{2} (θ) - 9 = 0

sin(a)+cos(a)=1

sin (a) + cos (a) = 1

2sin^2(x)+9sin(x)-5=0

2 sin^{2} (x) + 9 sin (x) - 5 = 0

10cos(2x)=0

10 cos (2 x) = 0

Frequently Asked Questions (FAQ)

What is the general solution for 2sech^2(x)+tanh(x)=0 ?
The general solution for 2sech^2(x)+tanh(x)=0 is x= 1/2 ln(-4+sqrt(17))