Precalculus Examples

${(x - 8)}^{2} = {(x + 4)}^{2}$

Step 1

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| x - 8 | = | x + 4 |$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the absolute value equation as four equations without absolute value bars.

$x - 8 = x + 4$

$x - 8 = - (x + 4)$

$- (x - 8) = x + 4$

$- (x - 8) = - (x + 4)$

Step 2.2

After simplifying, there are only two unique equations to be solved.

$x - 8 = x + 4$

$x - 8 = - (x + 4)$

Step 2.3

Solve $x - 8 = x + 4$ for $x$ .

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Step 2.3.1

Move all terms containing $x$ to the left side of the equation.

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Step 2.3.1.1

Subtract $x$ from both sides of the equation.

$x - 8 - x = 4$

Step 2.3.1.2

Combine the opposite terms in $x - 8 - x$ .

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Step 2.3.1.2.1

Subtract $x$ from $x$ .

$0 - 8 = 4$

Step 2.3.1.2.2

Subtract $8$ from $0$ .

$- 8 = 4$

Step 2.3.2

Since $- 8 \neq 4$ , there are no solutions.

No solution

Step 2.4

Solve $x - 8 = - (x + 4)$ for $x$ .

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Step 2.4.1

Simplify $- (x + 4)$ .

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Step 2.4.1.1

Rewrite.

$x - 8 = 0 + 0 - (x + 4)$

Step 2.4.1.2

Simplify by adding zeros.

$x - 8 = - (x + 4)$

Step 2.4.1.3

Apply the distributive property.

$x - 8 = - x - 1 \cdot 4$

Step 2.4.1.4

Multiply $- 1$ by $4$ .

$x - 8 = - x - 4$

Step 2.4.2

Move all terms containing $x$ to the left side of the equation.

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Step 2.4.2.1

Add $x$ to both sides of the equation.

$x - 8 + x = - 4$

Step 2.4.2.2

Add $x$ and $x$ .

$2 x - 8 = - 4$

Step 2.4.3

Move all terms not containing $x$ to the right side of the equation.

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Step 2.4.3.1

Add $8$ to both sides of the equation.

$2 x = - 4 + 8$

Step 2.4.3.2

Add $- 4$ and $8$ .

$2 x = 4$

Step 2.4.4

Divide each term in $2 x = 4$ by $2$ and simplify.

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Step 2.4.4.1

Divide each term in $2 x = 4$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $2$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2}$

Step 2.4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{2}$

Step 2.4.4.3

Simplify the right side.

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Step 2.4.4.3.1

Divide $4$ by $2$ .

$x = 2$

Step 2.5

List all of the solutions.

$x = 2$