What is the general solution for arcsin(6x)+arcsin(6sqrt(3)x)=-pi/2 ?

The general solution for arcsin(6x)+arcsin(6sqrt(3)x)=-pi/2 is x=-1/12

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arcsin(6x)+arcsin(6sqrt(3)x)=-pi/2

Solution

arcsin (6 x) + arcsin (63 x) = - \frac{π}{2}

Solution

x = - \frac{1}{12}

Solution steps

arcsin (6 x) + arcsin (63 x) = - \frac{π}{2}

Rewrite using trig identities

arcsin (6 x) + arcsin (63 x)

Use the Sum to Product identity:

arcsin (s) + arcsin (t) = arcsin (s 1 - t^{2} + t 1 - s^{2})

= arcsin (6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2})

arcsin (6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2}) = - \frac{π}{2}

Apply trig inverse properties

arcsin (6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2}) = - \frac{π}{2}

arcsin (x) = a \Rightarrow x = sin (a)

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = sin (- \frac{π}{2})

sin (- \frac{π}{2}) = - 1

sin (- \frac{π}{2})

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{π}{2}) = - sin (\frac{π}{2})

= - sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= - 1

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1

Solve

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1 : x = - \frac{1}{12}

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1

Remove square roots

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1

Subtract

63 x 1 - (6 x)^{2}

from both sides

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} - 63 x 1 - (6 x)^{2} = - 1 - 63 x 1 - (6 x)^{2}

Simplify

6 1 - (63 x)^{2} x = - 1 - 63 x 1 - (6 x)^{2}

Square both sides:

36 x^{2} - 3888 x^{4} = 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1

(6 1 - (63 x)^{2} x)^{2} = (- 1 - 63 x 1 - (6 x)^{2})^{2}

Expand

(6 1 - (63 x)^{2} x)^{2} : 36 x^{2} - 3888 x^{4}

(6 1 - (63 x)^{2} x)^{2}

Apply exponent rule:

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

= 6^{2} x^{2} (1 - (63 x)^{2})^{2}

(1 - (63 x)^{2})^{2} : 1 - (63 x)^{2}

Apply radical rule:

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

= ((1 - (63 x)^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (1 - (63 x)^{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

= 1 - (63 x)^{2}

= 6^{2} (1 - (63 x)^{2}) x^{2}

6^{2} = 36

= 36 (1 - (63 x)^{2}) x^{2}

Expand

36 (1 - (63 x)^{2}) x^{2} : 36 x^{2} - 3888 x^{4}

36 (1 - (63 x)^{2}) x^{2}

(63 x)^{2} = 6^{2} \cdot 3 x^{2}

(63 x)^{2}

Apply exponent rule:

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

= 6^{2} (3)^{2} x^{2}

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

= 6^{2} \cdot 3 x^{2}

= 36 x^{2} (- 6^{2} \cdot 3 x^{2} + 1)

6^{2} \cdot 3 x^{2} = 108 x^{2}

6^{2} \cdot 3 x^{2}

6^{2} = 36

= 36 \cdot 3 x^{2}

Multiply the numbers:

36 \cdot 3 = 108

= 108 x^{2}

= 36 x^{2} (- 108 x^{2} + 1)

= 36 x^{2} (1 - 108 x^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 36 x^{2}, b = 1, c = 108 x^{2}

= 36 x^{2} \cdot 1 - 36 x^{2} \cdot 108 x^{2}

= 36 \cdot 1 \cdot x^{2} - 36 \cdot 108 x^{2} x^{2}

Simplify

36 \cdot 1 \cdot x^{2} - 36 \cdot 108 x^{2} x^{2} : 36 x^{2} - 3888 x^{4}

36 \cdot 1 \cdot x^{2} - 36 \cdot 108 x^{2} x^{2}

36 \cdot 1 \cdot x^{2} = 36 x^{2}

36 \cdot 1 \cdot x^{2}

Multiply the numbers:

36 \cdot 1 = 36

= 36 x^{2}

36 \cdot 108 x^{2} x^{2} = 3888 x^{4}

36 \cdot 108 x^{2} x^{2}

Multiply the numbers:

36 \cdot 108 = 3888

= 3888 x^{2} x^{2}

Apply exponent rule:

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

x^{2} x^{2} = x^{2 + 2}

= 3888 x^{2 + 2}

Add the numbers:

2 + 2 = 4

= 3888 x^{4}

= 36 x^{2} - 3888 x^{4}

= 36 x^{2} - 3888 x^{4}

= 36 x^{2} - 3888 x^{4}

Expand

(- 1 - 63 x 1 - (6 x)^{2})^{2} : 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

(- 1 - 63 x 1 - (6 x)^{2})^{2}

= (- 1 - 63 1 - (6 x)^{2} x)^{2}

Apply Perfect Square Formula:

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

a = - 1, b = 63 x 1 - (6 x)^{2}

= (- 1)^{2} - 2 (- 1) \cdot 63 x 1 - (6 x)^{2} + (63 x 1 - (6 x)^{2})^{2}

Simplify

(- 1)^{2} - 2 (- 1) \cdot 63 x 1 - (6 x)^{2} + (63 x 1 - (6 x)^{2})^{2} : 1 + 123 1 - (6 x)^{2} x + 1081 - (6 x)^{2} x^{2}

(- 1)^{2} - 2 (- 1) \cdot 63 x 1 - (6 x)^{2} + (63 x 1 - (6 x)^{2})^{2}

Apply rule

- (- a) = a

= (- 1)^{2} + 2 \cdot 1 \cdot 63 x 1 - (6 x)^{2} + (63 x 1 - (6 x)^{2})^{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

2 \cdot 1 \cdot 63 x 1 - (6 x)^{2} = 123 1 - (6 x)^{2} x

2 \cdot 1 \cdot 63 x 1 - (6 x)^{2}

Multiply the numbers:

2 \cdot 1 \cdot 6 = 12

= 123 1 - (6 x)^{2} x

(63 x 1 - (6 x)^{2})^{2} = 1081 - (6 x)^{2} x^{2}

(63 x 1 - (6 x)^{2})^{2}

Apply exponent rule:

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

= 6^{2} (3)^{2} x^{2} (1 - (6 x)^{2})^{2}

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

= 6^{2} \cdot 3 x^{2} (1 - (6 x)^{2})^{2}

(1 - (6 x)^{2})^{2} : 1 - (6 x)^{2}

Apply radical rule:

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

= ((1 - (6 x)^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (1 - (6 x)^{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

= 1 - (6 x)^{2}

= 6^{2} \cdot 3 x^{2} (1 - (6 x)^{2})

Refine

= 108 (1 - (6 x)^{2}) x^{2}

= 1 + 123 1 - (6 x)^{2} x + 108 (1 - (6 x)^{2}) x^{2}

= 1 + 123 1 - (6 x)^{2} x + 108 (1 - (6 x)^{2}) x^{2}

Expand

1 + 123 1 - (6 x)^{2} x + 108 (1 - (6 x)^{2}) x^{2} : 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

1 + 123 1 - (6 x)^{2} x + 108 (1 - (6 x)^{2}) x^{2}

1 - (6 x)^{2} = 1 - 36 x^{2}

1 - (6 x)^{2}

(6 x)^{2} = 36 x^{2}

(6 x)^{2}

Apply exponent rule:

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

= 6^{2} x^{2}

6^{2} = 36

= 36 x^{2}

= 1 - 36 x^{2}

= 1 + 123 x - 36 x^{2} + 1 + 108 x^{2} (- (6 x)^{2} + 1)

(6 x)^{2} = 36 x^{2}

(6 x)^{2}

Apply exponent rule:

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

= 6^{2} x^{2}

6^{2} = 36

= 36 x^{2}

= 1 + 123 x - 36 x^{2} + 1 + 108 x^{2} (- 36 x^{2} + 1)

= 1 + 123 x 1 - 36 x^{2} + 108 x^{2} (1 - 36 x^{2})

Expand

108 x^{2} (1 - 36 x^{2}) : 108 x^{2} - 3888 x^{4}

108 x^{2} (1 - 36 x^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 108 x^{2}, b = 1, c = 36 x^{2}

= 108 x^{2} \cdot 1 - 108 x^{2} \cdot 36 x^{2}

= 108 \cdot 1 \cdot x^{2} - 108 \cdot 36 x^{2} x^{2}

Simplify

108 \cdot 1 \cdot x^{2} - 108 \cdot 36 x^{2} x^{2} : 108 x^{2} - 3888 x^{4}

108 \cdot 1 \cdot x^{2} - 108 \cdot 36 x^{2} x^{2}

108 \cdot 1 \cdot x^{2} = 108 x^{2}

108 \cdot 1 \cdot x^{2}

Multiply the numbers:

108 \cdot 1 = 108

= 108 x^{2}

108 \cdot 36 x^{2} x^{2} = 3888 x^{4}

108 \cdot 36 x^{2} x^{2}

Multiply the numbers:

108 \cdot 36 = 3888

= 3888 x^{2} x^{2}

Apply exponent rule:

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

x^{2} x^{2} = x^{2 + 2}

= 3888 x^{2 + 2}

Add the numbers:

2 + 2 = 4

= 3888 x^{4}

= 108 x^{2} - 3888 x^{4}

= 108 x^{2} - 3888 x^{4}

= 1 + 123 1 - 36 x^{2} x + 108 x^{2} - 3888 x^{4}

= 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

= 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

36 x^{2} - 3888 x^{4} = 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

36 x^{2} - 3888 x^{4} = 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

Subtract

108 x^{2} - 3888 x^{4}

from both sides

36 x^{2} - 3888 x^{4} - (108 x^{2} - 3888 x^{4}) = 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4} - (108 x^{2} - 3888 x^{4})

Simplify

- 72 x^{2} = 123 1 - 36 x^{2} x + 1

Subtract

1

from both sides

- 72 x^{2} - 1 = 123 1 - 36 x^{2} x + 1 - 1

Simplify

- 72 x^{2} - 1 = 123 1 - 36 x^{2} x

Square both sides:

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4}

36 x^{2} - 3888 x^{4} = 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

(- 72 x^{2} - 1)^{2} = (123 1 - 36 x^{2} x)^{2}

Expand

(- 72 x^{2} - 1)^{2} : 5184 x^{4} + 144 x^{2} + 1

(- 72 x^{2} - 1)^{2}

Apply Perfect Square Formula:

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

a = - 72 x^{2}, b = 1

= (- 72 x^{2})^{2} - 2 (- 72 x^{2}) \cdot 1 + 1^{2}

Simplify

(- 72 x^{2})^{2} - 2 (- 72 x^{2}) \cdot 1 + 1^{2} : 5184 x^{4} + 144 x^{2} + 1

(- 72 x^{2})^{2} - 2 (- 72 x^{2}) \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (- 72 x^{2})^{2} - 2 \cdot 1 \cdot (- 72 x^{2}) + 1

Apply rule

- (- a) = a

= (- 72 x^{2})^{2} + 2 \cdot 72 x^{2} \cdot 1 + 1

(- 72 x^{2})^{2} = 5184 x^{4}

(- 72 x^{2})^{2}

Apply exponent rule:

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

n

is even

(- 72 x^{2})^{2} = (72 x^{2})^{2}

= (72 x^{2})^{2}

Apply exponent rule:

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

= 7 2^{2} (x^{2})^{2}

(x^{2})^{2} : x^{4}

Apply exponent rule:

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

= x^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= x^{4}

= 7 2^{2} x^{4}

7 2^{2} = 5184

= 5184 x^{4}

2 \cdot 72 x^{2} \cdot 1 = 144 x^{2}

2 \cdot 72 x^{2} \cdot 1

Multiply the numbers:

2 \cdot 72 \cdot 1 = 144

= 144 x^{2}

= 5184 x^{4} + 144 x^{2} + 1

= 5184 x^{4} + 144 x^{2} + 1

Expand

(123 1 - 36 x^{2} x)^{2} : 432 x^{2} - 15552 x^{4}

(123 1 - 36 x^{2} x)^{2}

Apply exponent rule:

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

= 1 2^{2} (3)^{2} x^{2} (1 - 36 x^{2})^{2}

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

= 1 2^{2} \cdot 3 (1 - 36 x^{2})^{2} x^{2}

(1 - 36 x^{2})^{2} : 1 - 36 x^{2}

Apply radical rule:

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

= ((1 - 36 x^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (1 - 36 x^{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

= 1 - 36 x^{2}

= 1 2^{2} \cdot 3 (1 - 36 x^{2}) x^{2}

Refine

= 432 (1 - 36 x^{2}) x^{2}

Expand

432 (1 - 36 x^{2}) x^{2} : 432 x^{2} - 15552 x^{4}

432 (1 - 36 x^{2}) x^{2}

= 432 x^{2} (1 - 36 x^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 432 x^{2}, b = 1, c = 36 x^{2}

= 432 x^{2} \cdot 1 - 432 x^{2} \cdot 36 x^{2}

= 432 \cdot 1 \cdot x^{2} - 432 \cdot 36 x^{2} x^{2}

Simplify

432 \cdot 1 \cdot x^{2} - 432 \cdot 36 x^{2} x^{2} : 432 x^{2} - 15552 x^{4}

432 \cdot 1 \cdot x^{2} - 432 \cdot 36 x^{2} x^{2}

432 \cdot 1 \cdot x^{2} = 432 x^{2}

432 \cdot 1 \cdot x^{2}

Multiply the numbers:

432 \cdot 1 = 432

= 432 x^{2}

432 \cdot 36 x^{2} x^{2} = 15552 x^{4}

432 \cdot 36 x^{2} x^{2}

Multiply the numbers:

432 \cdot 36 = 15552

= 15552 x^{2} x^{2}

Apply exponent rule:

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

x^{2} x^{2} = x^{2 + 2}

= 15552 x^{2 + 2}

Add the numbers:

2 + 2 = 4

= 15552 x^{4}

= 432 x^{2} - 15552 x^{4}

= 432 x^{2} - 15552 x^{4}

= 432 x^{2} - 15552 x^{4}

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4}

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4}

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4}

Solve

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4} : x = \frac{1}{12}, x = - \frac{1}{12}

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4}

Move

15552 x^{4}

to the left side

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4}

Add

15552 x^{4}

to both sides

5184 x^{4} + 144 x^{2} + 1 + 15552 x^{4} = 432 x^{2} - 15552 x^{4} + 15552 x^{4}

Simplify

20736 x^{4} + 144 x^{2} + 1 = 432 x^{2}

20736 x^{4} + 144 x^{2} + 1 = 432 x^{2}

Move

432 x^{2}

to the left side

20736 x^{4} + 144 x^{2} + 1 = 432 x^{2}

Subtract

432 x^{2}

from both sides

20736 x^{4} + 144 x^{2} + 1 - 432 x^{2} = 432 x^{2} - 432 x^{2}

Simplify

20736 x^{4} - 288 x^{2} + 1 = 0

20736 x^{4} - 288 x^{2} + 1 = 0

Divide both sides by

20736

\frac{20736 x ^{4}}{20736} - \frac{288 x ^{2}}{20736} + \frac{1}{20736} = \frac{0}{20736}

Write in the standard form

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

x^{4} - \frac{x ^{2}}{72} + \frac{1}{20736} = 0

Rewrite the equation with

u = x^{2}

and

u^{2} = x^{4}

u^{2} - \frac{u}{72} + \frac{1}{20736} = 0

Solve

u^{2} - \frac{u}{72} + \frac{1}{20736} = 0 : u = \frac{1}{144}

u^{2} - \frac{u}{72} + \frac{1}{20736} = 0

Find Least Common Multiplier of

72, 20736 : 20736

72, 20736

Least Common Multiplier (LCM)

Prime factorization of

72 : 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

72

72

divides by

272 = 36 \cdot 2

= 2 \cdot 36

36

divides by

236 = 18 \cdot 2

= 2 \cdot 2 \cdot 18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

Prime factorization of

20736 : 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3

20736

20736

divides by

220736 = 10368 \cdot 2

= 2 \cdot 10368

10368

divides by

210368 = 5184 \cdot 2

= 2 \cdot 2 \cdot 5184

5184

divides by

25184 = 2592 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2592

2592

divides by

22592 = 1296 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 1296

1296

divides by

21296 = 648 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 648

648

divides by

2648 = 324 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 324

324

divides by

2324 = 162 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 162

162

divides by

2162 = 81 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 81

81

divides by

381 = 27 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 27

27

divides by

327 = 9 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3

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

72

20736

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 20736

= 20736

Multiply by LCM=

20736

u^{2} \cdot 20736 - \frac{u}{72} \cdot 20736 + \frac{1}{20736} \cdot 20736 = 0 \cdot 20736

Simplify

20736 u^{2} - 288 u + 1 = 0

Divide both sides by

20736

\frac{20736 u ^{2}}{20736} - \frac{288 u}{20736} + \frac{1}{20736} = \frac{0}{20736}

Write in the standard form

a x^{2} + b x + c = 0

u^{2} - \frac{u}{72} + \frac{1}{20736} = 0

Solve with the quadratic formula

u^{2} - \frac{u}{72} + \frac{1}{20736} = 0

Quadratic Equation Formula:

For

a = 1, b = - \frac{1}{72}, c = \frac{1}{20736}

u_{1, 2} = \frac{- ( - \frac{1}{72} ) \pm ( - \frac{1}{72} ) ^{2} - 4 \cdot 1 \cdot \frac{1}{20736}}{2 \cdot 1}

u_{1, 2} = \frac{- ( - \frac{1}{72} ) \pm ( - \frac{1}{72} ) ^{2} - 4 \cdot 1 \cdot \frac{1}{20736}}{2 \cdot 1}

(- \frac{1}{72})^{2} - 4 \cdot 1 \cdot \frac{1}{20736} = 0

(- \frac{1}{72})^{2} - 4 \cdot 1 \cdot \frac{1}{20736}

(- \frac{1}{72})^{2} = \frac{1}{7 2 ^{2}}

(- \frac{1}{72})^{2}

Apply exponent rule:

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

n

is even

(- \frac{1}{72})^{2} = (\frac{1}{72})^{2}

= (\frac{1}{72})^{2}

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

4 \cdot 1 \cdot \frac{1}{20736} = \frac{1}{5184}

4 \cdot 1 \cdot \frac{1}{20736}

Multiply fractions:

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

= 1 \cdot \frac{1 \cdot 4}{20736}

\frac{1 \cdot 4}{20736} = \frac{1}{5184}

\frac{1 \cdot 4}{20736}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{20736}

Cancel the common factor:

4

= \frac{1}{5184}

= 1 \cdot \frac{1}{5184}

Multiply:

1 \cdot \frac{1}{5184} = \frac{1}{5184}

= \frac{1}{5184}

= \frac{1}{7 2 ^{2}} - \frac{1}{5184}

7 2^{2} = 5184

= \frac{1}{5184} - \frac{1}{5184}

Add similar elements:

\frac{1}{5184} - \frac{1}{5184} = 0

= 0

u_{1, 2} = \frac{- ( - \frac{1}{72} ) \pm 0}{2 \cdot 1}

u = \frac{- ( - \frac{1}{72} )}{2 \cdot 1}

\frac{- ( - \frac{1}{72} )}{2 \cdot 1} = \frac{1}{144}

\frac{- ( - \frac{1}{72} )}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{\frac{1}{72}}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{\frac{1}{72}}{2}

Apply the fraction rule:

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

= \frac{1}{72 \cdot 2}

Multiply the numbers:

72 \cdot 2 = 144

= \frac{1}{144}

u = \frac{1}{144}

The solution to the quadratic equation is:

u = \frac{1}{144}

u = \frac{1}{144}

Substitute back

u = x^{2},

solve for

x

Solve

x^{2} = \frac{1}{144} : x = \frac{1}{12}, x = - \frac{1}{12}

x^{2} = \frac{1}{144}

For

x^{2} = f (a)

the solutions are

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

x = \frac{1}{144}, x = - \frac{1}{144}

\frac{1}{144} = \frac{1}{12}

\frac{1}{144}

Apply radical rule:

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

= \frac{1}{144}

Apply radical rule:

1 = 1

1 = 1

= \frac{1}{144}

144 = 12

144

Factor the number:

144 = 1 2^{2}

= 1 2^{2}

Apply radical rule:

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

1 2^{2} = 12

= 12

= \frac{1}{12}

- \frac{1}{144} = - \frac{1}{12}

- \frac{1}{144}

Apply radical rule:

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

= - \frac{1}{144}

Apply radical rule:

1 = 1

1 = 1

= - \frac{1}{144}

144 = 12

144

Factor the number:

144 = 1 2^{2}

= 1 2^{2}

Apply radical rule:

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

1 2^{2} = 12

= 12

= - \frac{1}{12}

x = \frac{1}{12}, x = - \frac{1}{12}

The solutions are

x = \frac{1}{12}, x = - \frac{1}{12}

x = \frac{1}{12}, x = - \frac{1}{12}

Verify Solutions:

x = \frac{1}{12}

False

, x = - \frac{1}{12}

True

Check the solutions by plugging them into

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1

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

Plug in

x = \frac{1}{12} :

False

6 (\frac{1}{12}) 1 - (63 (\frac{1}{12}))^{2} + 63 (\frac{1}{12}) 1 - (6 (\frac{1}{12}))^{2} = - 1

6 (\frac{1}{12}) 1 - (63 (\frac{1}{12}))^{2} + 63 (\frac{1}{12}) 1 - (6 (\frac{1}{12}))^{2} = 1

6 (\frac{1}{12}) 1 - (63 (\frac{1}{12}))^{2} + 63 (\frac{1}{12}) 1 - (6 (\frac{1}{12}))^{2}

Remove parentheses:

(a) = a

= 6 \cdot \frac{1}{12} 1 - (63 \frac{1}{12})^{2} + 63 \frac{1}{12} 1 - (6 \cdot \frac{1}{12})^{2}

6 \cdot \frac{1}{12} 1 - (63 \frac{1}{12})^{2} = \frac{1}{4}

6 \cdot \frac{1}{12} 1 - (63 \frac{1}{12})^{2}

1 - (63 \frac{1}{12})^{2} = \frac{1}{2}

1 - (63 \frac{1}{12})^{2}

(63 \frac{1}{12})^{2} = \frac{3}{4}

(63 \frac{1}{12})^{2}

Multiply

63 \frac{1}{12} : \frac{3}{2}

63 \frac{1}{12}

Multiply fractions:

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

= \frac{1 \cdot 6 3}{12}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6 3}{12}

Cancel the common factor:

6

= \frac{3}{2}

= (\frac{3}{2})^{2}

Apply exponent rule:

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

= \frac{( 3 ) ^{2}}{2 ^{2}}

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

= \frac{3}{2 ^{2}}

2^{2} = 4

= \frac{3}{4}

= 1 - \frac{3}{4}

Join

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

1 - \frac{3}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{3}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 4 - 3}{4}

1 \cdot 4 - 3 = 1

1 \cdot 4 - 3

Multiply the numbers:

1 \cdot 4 = 4

= 4 - 3

Subtract the numbers:

4 - 3 = 1

= 1

= \frac{1}{4}

= \frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

= 6 \cdot \frac{1}{2} \cdot \frac{1}{12}

Multiply fractions:

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

= \frac{1 \cdot 1 \cdot 6}{12 \cdot 2}

Multiply the numbers:

1 \cdot 1 \cdot 6 = 6

= \frac{6}{12 \cdot 2}

Multiply the numbers:

12 \cdot 2 = 24

= \frac{6}{24}

Cancel the common factor:

6

= \frac{1}{4}

63 \frac{1}{12} 1 - (6 \cdot \frac{1}{12})^{2} = \frac{3}{4}

63 \frac{1}{12} 1 - (6 \cdot \frac{1}{12})^{2}

1 - (6 \cdot \frac{1}{12})^{2} = \frac{3}{2}

1 - (6 \cdot \frac{1}{12})^{2}

(6 \cdot \frac{1}{12})^{2} = \frac{1}{4}

(6 \cdot \frac{1}{12})^{2}

Multiply

6 \cdot \frac{1}{12} : \frac{1}{2}

6 \cdot \frac{1}{12}

Multiply fractions:

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

= \frac{1 \cdot 6}{12}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6}{12}

Cancel the common factor:

6

= \frac{1}{2}

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

2^{2} = 4

= \frac{1}{4}

= 1 - \frac{1}{4}

Join

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

1 - \frac{1}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 4 - 1 = 3

1 \cdot 4 - 1

Multiply the numbers:

1 \cdot 4 = 4

= 4 - 1

Subtract the numbers:

4 - 1 = 3

= 3

= \frac{3}{4}

= \frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

= 63 \frac{1}{12} \cdot \frac{3}{2}

Multiply fractions:

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

= \frac{1 \cdot 3 \cdot 6 3}{12 \cdot 2}

1 \cdot 3 \cdot 63 = 18

1 \cdot 3 \cdot 63

Multiply the numbers:

1 \cdot 6 = 6

= 633

Apply radical rule:

a a = a

33 = 3

= 6 \cdot 3

Multiply the numbers:

6 \cdot 3 = 18

= 18

= \frac{18}{12 \cdot 2}

Multiply the numbers:

12 \cdot 2 = 24

= \frac{18}{24}

Cancel the common factor:

6

= \frac{3}{4}

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

Apply rule

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

= \frac{1 + 3}{4}

Add the numbers:

1 + 3 = 4

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

1 = - 1

False

Plug in

x = - \frac{1}{12} :

True

6 (- \frac{1}{12}) 1 - (63 (- \frac{1}{12}))^{2} + 63 (- \frac{1}{12}) 1 - (6 (- \frac{1}{12}))^{2} = - 1

6 (- \frac{1}{12}) 1 - (63 (- \frac{1}{12}))^{2} + 63 (- \frac{1}{12}) 1 - (6 (- \frac{1}{12}))^{2} = - 1

6 (- \frac{1}{12}) 1 - (63 (- \frac{1}{12}))^{2} + 63 (- \frac{1}{12}) 1 - (6 (- \frac{1}{12}))^{2}

Remove parentheses:

(- a) = - a

= - 6 \cdot \frac{1}{12} 1 - (- 63 \frac{1}{12})^{2} - 63 \frac{1}{12} 1 - (- 6 \cdot \frac{1}{12})^{2}

6 \cdot \frac{1}{12} 1 - (- 63 \frac{1}{12})^{2} = \frac{1}{4}

6 \cdot \frac{1}{12} 1 - (- 63 \frac{1}{12})^{2}

1 - (- 63 \frac{1}{12})^{2} = \frac{1}{2}

1 - (- 63 \frac{1}{12})^{2}

(- 63 \frac{1}{12})^{2} = \frac{3}{4}

(- 63 \frac{1}{12})^{2}

Multiply

- 63 \frac{1}{12} : - \frac{3}{2}

- 63 \frac{1}{12}

Multiply fractions:

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

= - \frac{1 \cdot 6 3}{12}

Multiply the numbers:

1 \cdot 6 = 6

= - \frac{6 3}{12}

Cancel the common factor:

6

= - \frac{3}{2}

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

Apply exponent rule:

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

n

is even

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

= (\frac{3}{2})^{2}

Apply exponent rule:

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

= \frac{( 3 ) ^{2}}{2 ^{2}}

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\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

= 3

= \frac{3}{2 ^{2}}

2^{2} = 4

= \frac{3}{4}

= 1 - \frac{3}{4}

Join

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

1 - \frac{3}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{3}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 4 - 3}{4}

1 \cdot 4 - 3 = 1

1 \cdot 4 - 3

Multiply the numbers:

1 \cdot 4 = 4

= 4 - 3

Subtract the numbers:

4 - 3 = 1

= 1

= \frac{1}{4}

= \frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

= 6 \cdot \frac{1}{2} \cdot \frac{1}{12}

Multiply fractions:

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

= \frac{1 \cdot 1 \cdot 6}{12 \cdot 2}

Multiply the numbers:

1 \cdot 1 \cdot 6 = 6

= \frac{6}{12 \cdot 2}

Multiply the numbers:

12 \cdot 2 = 24

= \frac{6}{24}

Cancel the common factor:

6

= \frac{1}{4}

63 \frac{1}{12} 1 - (- 6 \cdot \frac{1}{12})^{2} = \frac{3}{4}

63 \frac{1}{12} 1 - (- 6 \cdot \frac{1}{12})^{2}

1 - (- 6 \cdot \frac{1}{12})^{2} = \frac{3}{2}

1 - (- 6 \cdot \frac{1}{12})^{2}

(- 6 \cdot \frac{1}{12})^{2} = \frac{1}{4}

(- 6 \cdot \frac{1}{12})^{2}

Multiply

- 6 \cdot \frac{1}{12} : - \frac{1}{2}

- 6 \cdot \frac{1}{12}

Multiply fractions:

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

= - \frac{1 \cdot 6}{12}

Multiply the numbers:

1 \cdot 6 = 6

= - \frac{6}{12}

Cancel the common factor:

6

= - \frac{1}{2}

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

Apply exponent rule:

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

n

is even

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

2^{2} = 4

= \frac{1}{4}

= 1 - \frac{1}{4}

Join

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

1 - \frac{1}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 4 - 1 = 3

1 \cdot 4 - 1

Multiply the numbers:

1 \cdot 4 = 4

= 4 - 1

Subtract the numbers:

4 - 1 = 3

= 3

= \frac{3}{4}

= \frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

= 63 \frac{1}{12} \cdot \frac{3}{2}

Multiply fractions:

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

= \frac{1 \cdot 3 \cdot 6 3}{12 \cdot 2}

1 \cdot 3 \cdot 63 = 18

1 \cdot 3 \cdot 63

Multiply the numbers:

1 \cdot 6 = 6

= 633

Apply radical rule:

a a = a

33 = 3

= 6 \cdot 3

Multiply the numbers:

6 \cdot 3 = 18

= 18

= \frac{18}{12 \cdot 2}

Multiply the numbers:

12 \cdot 2 = 24

= \frac{18}{24}

Cancel the common factor:

6

= \frac{3}{4}

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

Apply rule

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

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

Subtract the numbers:

- 1 - 3 = - 4

= \frac{- 4}{4}

Apply the fraction rule:

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

= - \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= - 1

- 1 = - 1

True

The solution is

x = - \frac{1}{12}

x = - \frac{1}{12}

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

arcsin (6 x) + arcsin (63 x) = - \frac{π}{2}

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

Check the solution

- \frac{1}{12} :

True

- \frac{1}{12}

Plug in

n = 1

- \frac{1}{12}

For

arcsin (6 x) + arcsin (63 x) = - \frac{π}{2}

plug in

x = - \frac{1}{12}

arcsin (6 (- \frac{1}{12})) + arcsin (63 (- \frac{1}{12})) = - \frac{π}{2}

Refine

- 1.57079 \dots = - 1.57079 \dots

\Rightarrow True

x = - \frac{1}{12}

Popular Examples

cos(x)= 1/2 ,0<= x<= 360

cos (x) = \frac{1}{2}, 0^{\circ} \leq x \leq 36 0^{\circ}

-44sin(2x)+42cos(2x)=0

- 44 sin (2 x) + 42 cos (2 x) = 0

cos^2(x)=cos(x)+2

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

cos^2(x)=cos(x)-1

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

4+4sin(θ)= 3/(1-sin(θ))

4 + 4 sin (θ) = \frac{3}{1 - sin ( θ )}

Frequently Asked Questions (FAQ)

What is the general solution for arcsin(6x)+arcsin(6sqrt(3)x)=-pi/2 ?
The general solution for arcsin(6x)+arcsin(6sqrt(3)x)=-pi/2 is x=-1/12