Precalculus Examples

${(x - 9)}^{2} = {(x + 3)}^{2}$

Step 1

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

$| x - 9 | = | x + 3 |$

Step 2

Solve for $x$ .

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Step 2.1

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

$x - 9 = x + 3$

$x - 9 = - (x + 3)$

$- (x - 9) = x + 3$

$- (x - 9) = - (x + 3)$

Step 2.2

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

$x - 9 = x + 3$

$x - 9 = - (x + 3)$

Step 2.3

Solve $x - 9 = x + 3$ 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.

$x - 9 - x = 3$

Step 2.3.1.2

Combine the opposite terms in $x - 9 - x$ .

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Step 2.3.1.2.1

Subtract $x$ from $x$ .

$0 - 9 = 3$

Step 2.3.1.2.2

Subtract $9$ from $0$ .

$- 9 = 3$

Step 2.3.2

Since $- 9 \neq 3$ , there are no solutions.

No solution

Step 2.4

Solve $x - 9 = - (x + 3)$ for $x$ .

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Step 2.4.1

Simplify $- (x + 3)$ .

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Step 2.4.1.1

Rewrite.

$x - 9 = 0 + 0 - (x + 3)$

Step 2.4.1.2

Simplify by adding zeros.

$x - 9 = - (x + 3)$

Step 2.4.1.3

Apply the distributive property.

$x - 9 = - x - 1 \cdot 3$

Step 2.4.1.4

Multiply $- 1$ by $3$ .

$x - 9 = - x - 3$

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.

$x - 9 + x = - 3$

Step 2.4.2.2

Add $x$ and $x$ .

$2 x - 9 = - 3$

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 $9$ to both sides of the equation.

$2 x = - 3 + 9$

Step 2.4.3.2

Add $- 3$ and $9$ .

$2 x = 6$

Step 2.4.4

Divide each term in $2 x = 6$ by $2$ and simplify.

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Step 2.4.4.1

Divide each term in $2 x = 6$ by $2$ .

$\frac{2 x}{2} = \frac{6}{2}$

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.

$\frac{2 x}{2} = \frac{6}{2}$

Step 2.4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{2}$

Step 2.4.4.3

Simplify the right side.

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Step 2.4.4.3.1

Divide $6$ by $2$ .

$x = 3$

Step 2.5

List all of the solutions.

$x = 3$