What is the general solution for arccos(1-x)+arccos(x)=arccos(-x) ?

The general solution for arccos(1-x)+arccos(x)=arccos(-x) is x=0,x= 1/2

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arccos(1-x)+arccos(x)=arccos(-x)

Solution

arccos (1 - x) + arccos (x) = arccos (- x)

Solution

x = 0, x = \frac{1}{2}

Solution steps

arccos (1 - x) + arccos (x) = arccos (- x)

a = b \Rightarrow cos (a) = cos (b)

cos (arccos (1 - x) + arccos (x)) = cos (arccos (- x))

Use the following identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

cos (arccos (1 - x)) cos (arccos (x)) - sin (arccos (1 - x)) sin (arccos (x)) = cos (arccos (- x))

Use the following identity:

cos (arccos (x)) = x

Use the following identity:

cos (arccos (x)) = x

Use the following identity:

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

Use the following identity:

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

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

Solve

(1 - x) x - 1 - (1 - x)^{2} 1 - x^{2} = - x : x = 0, x = \frac{1}{2}

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

Expand

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

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

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

Expand

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

x (1 - x)

Apply the distributive law:

a (b - c) = ab - a c

a = x, b = 1, c = x

= x \cdot 1 - xx

= 1 \cdot x - xx

Simplify

1 \cdot x - xx : x - x^{2}

1 \cdot x - xx

1 \cdot x = x

1 \cdot x

Multiply:

1 \cdot x = x

= x

xx = x^{2}

xx

Apply exponent rule:

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

xx = x^{1 + 1}

= x^{1 + 1}

Add the numbers:

1 + 1 = 2

= x^{2}

= x - x^{2}

= x - x^{2}

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

Expand

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

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

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

1 - (1 - x)^{2}

Expand

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

1 - (1 - x)^{2}

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

Apply Perfect Square Formula:

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

a = 1, b = x

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

Simplify

1^{2} - 2 \cdot 1 \cdot x + x^{2} : 1 - 2 x + x^{2}

1^{2} - 2 \cdot 1 \cdot x + x^{2}

Apply rule

1^{a} = 1

1^{2} = 1

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

Multiply the numbers:

2 \cdot 1 = 2

= 1 - 2 x + x^{2}

= 1 - 2 x + x^{2}

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

= - 1 + 2 x - x^{2}

= 1 - 1 + 2 x - x^{2}

1 - 1 = 0

= - x^{2} + 2 x

= - x^{2} + 2 x

= x - x^{2} - - x^{2} + 2 x - x^{2} + 1

= x - x^{2} - - x^{2} + 2 x 1 - x^{2}

x - x^{2} - - x^{2} + 2 x 1 - x^{2} = - x

Remove square roots

x - x^{2} - - x^{2} + 2 x 1 - x^{2} = - x

Subtract

x - x^{2}

from both sides

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

Simplify

- - x^{2} + 2 x 1 - x^{2} = - 2 x + x^{2}

Square both sides:

- x^{2} + x^{4} + 2 x - 2 x^{3} = 4 x^{2} - 4 x^{3} + x^{4}

x - x^{2} - - x^{2} + 2 x 1 - x^{2} = - x

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

Expand

(- - x^{2} + 2 x 1 - x^{2})^{2} : - x^{2} + x^{4} + 2 x - 2 x^{3}

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

Apply exponent rule:

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

n

is even

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

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

Apply exponent rule:

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

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

(- x^{2} + 2 x)^{2} : - x^{2} + 2 x

Apply radical rule:

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

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

Apply exponent rule:

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

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

= - x^{2} + 2 x

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

= (1 - 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 - x^{2}

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

Expand

(- x^{2} + 2 x) (1 - x^{2}) : - x^{2} + x^{4} + 2 x - 2 x^{3}

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

Apply FOIL method:

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

a = - x^{2}, b = 2 x, c = 1, d = - x^{2}

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

Apply minus-plus rules

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

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

Simplify

- 1 \cdot x^{2} + x^{2} x^{2} + 2 \cdot 1 \cdot x - 2 x^{2} x : - x^{2} + x^{4} + 2 x - 2 x^{3}

- 1 \cdot x^{2} + x^{2} x^{2} + 2 \cdot 1 \cdot x - 2 x^{2} x

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

1 \cdot x^{2}

Multiply:

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

= x^{2}

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

x^{2} x^{2}

Apply exponent rule:

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

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

= x^{2 + 2}

Add the numbers:

2 + 2 = 4

= x^{4}

2 \cdot 1 \cdot x = 2 x

2 \cdot 1 \cdot x

Multiply the numbers:

2 \cdot 1 = 2

= 2 x

2 x^{2} x = 2 x^{3}

2 x^{2} x

Apply exponent rule:

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

x^{2} x = x^{2 + 1}

= 2 x^{2 + 1}

Add the numbers:

2 + 1 = 3

= 2 x^{3}

= - x^{2} + x^{4} + 2 x - 2 x^{3}

= - x^{2} + x^{4} + 2 x - 2 x^{3}

= - x^{2} + x^{4} + 2 x - 2 x^{3}

Expand

(- 2 x + x^{2})^{2} : 4 x^{2} - 4 x^{3} + x^{4}

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

Apply Perfect Square Formula:

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

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

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

Simplify

(- 2 x)^{2} + 2 (- 2 x) x^{2} + (x^{2})^{2} : 4 x^{2} - 4 x^{3} + x^{4}

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

Remove parentheses:

(- a) = - a

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

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

(- 2 x)^{2}

Apply exponent rule:

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

n

is even

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

= (2 x)^{2}

Apply exponent rule:

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

= 2^{2} x^{2}

2^{2} = 4

= 4 x^{2}

2 \cdot 2 x x^{2} = 4 x^{3}

2 \cdot 2 x x^{2}

Multiply the numbers:

2 \cdot 2 = 4

= 4 x^{2} x

Apply exponent rule:

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

x x^{2} = x^{1 + 2}

= 4 x^{1 + 2}

Add the numbers:

1 + 2 = 3

= 4 x^{3}

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

(x^{2})^{2}

Apply exponent rule:

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

= x^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= x^{4}

= 4 x^{2} - 4 x^{3} + x^{4}

= 4 x^{2} - 4 x^{3} + x^{4}

- x^{2} + x^{4} + 2 x - 2 x^{3} = 4 x^{2} - 4 x^{3} + x^{4}

- x^{2} + x^{4} + 2 x - 2 x^{3} = 4 x^{2} - 4 x^{3} + x^{4}

- x^{2} + x^{4} + 2 x - 2 x^{3} = 4 x^{2} - 4 x^{3} + x^{4}

Solve

- x^{2} + x^{4} + 2 x - 2 x^{3} = 4 x^{2} - 4 x^{3} + x^{4} : x = 0, x = \frac{1}{2}, x = 2

- x^{2} + x^{4} + 2 x - 2 x^{3} = 4 x^{2} - 4 x^{3} + x^{4}

Subtract

4 x^{2} - 4 x^{3} + x^{4}

from both sides

- x^{2} + x^{4} + 2 x - 2 x^{3} - (4 x^{2} - 4 x^{3} + x^{4}) = 4 x^{2} - 4 x^{3} + x^{4} - (4 x^{2} - 4 x^{3} + x^{4})

Simplify

2 x^{3} - 5 x^{2} + 2 x = 0

Factor

2 x^{3} - 5 x^{2} + 2 x : x (2 x - 1) (x - 2)

2 x^{3} - 5 x^{2} + 2 x

Factor out common term

x : x (2 x^{2} - 5 x + 2)

2 x^{3} - 5 x^{2} + 2 x

Apply exponent rule:

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

x^{2} = xx

= 2 x^{2} x - 5 xx + 2 x

Factor out common term

x

= x (2 x^{2} - 5 x + 2)

= x (2 x^{2} - 5 x + 2)

Factor

2 x^{2} - 5 x + 2 : (2 x - 1) (x - 2)

2 x^{2} - 5 x + 2

Break the expression into groups

2 x^{2} - 5 x + 2

Definition

Factors of

4 : 1, 2, 4

4

Divisors (Factors)

Find the Prime factors of

4 : 2, 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2

Add the prime factors:

2

Add 1 and the number

4

itself

1, 4

The factors of

4

1, 2, 4

Negative factors of

4 : - 1, - 2, - 4

Multiply the factors by

- 1

to get the negative factors

- 1, - 2, - 4

For every two factors such that

u * v = 4,

check if

u + v = - 5

Check

u = 1, v = 4 : u * v = 4, u + v = 5 \Rightarrow

FalseCheck

u = 2, v = 2 : u * v = 4, u + v = 4 \Rightarrow

False

u = - 1, v = - 4

Group into

(a x^{2} + ux) + (vx + c)

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

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

Factor out

x

from

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

2 x^{2} - x

Apply exponent rule:

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

x^{2} = xx

= 2 xx - x

Factor out common term

x

= x (2 x - 1)

Factor out

- 2

from

- 4 x + 2 : - 2 (2 x - 1)

- 4 x + 2

Rewrite

4

2 \cdot 2

= - 2 \cdot 2 x + 2

Factor out common term

- 2

= - 2 (2 x - 1)

= x (2 x - 1) - 2 (2 x - 1)

Factor out common term

2 x - 1

= (2 x - 1) (x - 2)

= x (2 x - 1) (x - 2)

x (2 x - 1) (x - 2) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

x = 0 or 2 x - 1 = 0 or x - 2 = 0

Solve

2 x - 1 = 0 : x = \frac{1}{2}

2 x - 1 = 0

Move

1

to the right side

2 x - 1 = 0

Add

1

to both sides

2 x - 1 + 1 = 0 + 1

Simplify

2 x = 1

2 x = 1

Divide both sides by

2

2 x = 1

Divide both sides by

2

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

Simplify

x = \frac{1}{2}

x = \frac{1}{2}

Solve

x - 2 = 0 : x = 2

x - 2 = 0

Move

2

to the right side

x - 2 = 0

Add

2

to both sides

x - 2 + 2 = 0 + 2

Simplify

x = 2

x = 2

The solutions are

x = 0, x = \frac{1}{2}, x = 2

x = 0, x = \frac{1}{2}, x = 2

Verify Solutions:

x = 0

True

, x = \frac{1}{2}

True

, x = 2

False

Check the solutions by plugging them into

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

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

Plug in

x = 0 :

True

(1 - 0) \cdot 0 - 1 - (1 - 0)^{2} 1 - 0^{2} = - 0

(1 - 0) \cdot 0 - 1 - (1 - 0)^{2} 1 - 0^{2} = 0

(1 - 0) \cdot 0 - 1 - (1 - 0)^{2} 1 - 0^{2}

Apply rule

0^{a} = 0

0^{2} = 0

= 0 \cdot (1 - 0) - - (1 - 0)^{2} + 1 1 - 0

(1 - 0) \cdot 0 = 0

(1 - 0) \cdot 0

Subtract the numbers:

1 - 0 = 1

= 1 \cdot 0

Apply rule

0 \cdot a = 0

= 0

1 - (1 - 0)^{2} 1 - 0 = 0

1 - (1 - 0)^{2} 1 - 0

1 - (1 - 0)^{2} = 0

1 - (1 - 0)^{2}

(1 - 0)^{2} = 1

(1 - 0)^{2}

Subtract the numbers:

1 - 0 = 1

= 1^{2}

Apply rule

1^{a} = 1

= 1

= 1 - 1

Subtract the numbers:

1 - 1 = 0

= 0

Apply rule

0 = 0

= 0

= 0 \cdot 1 - 0

1 - 0 = 1

1 - 0

Subtract the numbers:

1 - 0 = 1

= 1

Apply rule

1 = 1

= 1

= 0 \cdot 1

Apply rule

0 \cdot a = 0

= 0

= 0 - 0

Subtract the numbers:

0 - 0 = 0

= 0

- 0 = 0

- 0

= 0

0 = 0

True

Plug in

x = \frac{1}{2} :

True

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

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

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

Remove parentheses:

(a) = a

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

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

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

Join

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

1 - \frac{1}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 2 - 1 = 1

1 \cdot 2 - 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= \frac{1}{2}

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

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

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

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

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

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

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

Join

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

1 - \frac{1}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 2 - 1 = 1

1 \cdot 2 - 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= \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}

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

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

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

(\frac{1}{2})^{2} = \frac{1}{4}

(\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}

= \frac{3}{2} \cdot \frac{3}{2}

Multiply fractions:

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

= \frac{3 3}{2 \cdot 2}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \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 = - 2

= \frac{- 2}{4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

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

- (\frac{1}{2})

Remove parentheses:

(a) = a

= - \frac{1}{2}

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

True

Plug in

x = 2 :

False

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

Simplify

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

Undefined

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

(1 - 2) \cdot 2 = - 2

(1 - 2) \cdot 2

Subtract the numbers:

1 - 2 = - 1

= (- 1) \cdot 2

Remove parentheses:

(- a) = - a

= - 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= - 2

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

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

Undefined

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

1 - (1 - 2)^{2} = 0

1 - (1 - 2)^{2}

(1 - 2)^{2} = 1

(1 - 2)^{2}

Subtract the numbers:

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

= 1 - 1

Subtract the numbers:

1 - 1 = 0

= 0

Apply rule

0 = 0

= 0

= 0 \cdot 1 - 2^{2}

1 - 2^{2} = - 3

1 - 2^{2}

2^{2} = 4

= 1 - 4

Subtract the numbers:

1 - 4 = - 3

= - 3

= 0 \cdot - 3

a, a < 0

is undefined

= Undefined

= Undefined

Undefined

= - 2

False

The solutions are

x = 0, x = \frac{1}{2}

x = 0, x = \frac{1}{2}

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

arccos (1 - x) + arccos (x) = arccos (- x)

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

Check the solution

0 :

True

0

Plug in

n = 1

0

For

arccos (1 - x) + arccos (x) = arccos (- x)

plug in

x = 0

arccos (1 - 0) + arccos (0) = arccos (- 0)

Refine

1.57079 \dots = 1.57079 \dots

\Rightarrow True

Check the solution

\frac{1}{2} :

True

\frac{1}{2}

Plug in

n = 1

\frac{1}{2}

For

arccos (1 - x) + arccos (x) = arccos (- x)

plug in

x = \frac{1}{2}

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

Refine

2.09439 \dots = 2.09439 \dots

\Rightarrow True

x = 0, x = \frac{1}{2}

Popular Examples

tan(x)=0.75

tan (x) = 0.75

2sin(x)tan(x)=tan(x)

2 sin (x) tan (x) = tan (x)

4csc(θ)+5=0

4 csc (θ) + 5 = 0

cos(x)=-cos(x)+2

cos (x) = - cos (x) + 2

4cos^2(2x)-3=0

4 cos^{2} (2 x) - 3 = 0

Frequently Asked Questions (FAQ)

What is the general solution for arccos(1-x)+arccos(x)=arccos(-x) ?
The general solution for arccos(1-x)+arccos(x)=arccos(-x) is x=0,x= 1/2