Precalculus Examples

${(x - 7)}^{2} = {(x + 3)}^{2}$

Step 1

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

$| x - 7 | = | x + 3 |$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

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

$x - 7 = x + 3$

$x - 7 = - (x + 3)$

$- (x - 7) = x + 3$

$- (x - 7) = - (x + 3)$

Step 2.2

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

$x - 7 = x + 3$

$x - 7 = - (x + 3)$

Step 2.3

Solve $x - 7 = x + 3$ for $x$ .

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

Subtract $x$ from both sides of the equation.

$x - 7 - x = 3$

Step 2.3.1.2

Combine the opposite terms in $x - 7 - x$ .

Tap for more steps...

Step 2.3.1.2.1

Subtract $x$ from $x$ .

$0 - 7 = 3$

Step 2.3.1.2.2

Subtract $7$ from $0$ .

$- 7 = 3$

Step 2.3.2

Since $- 7 \neq 3$ , there are no solutions.

No solution

Step 2.4

Solve $x - 7 = - (x + 3)$ for $x$ .

Tap for more steps...

Step 2.4.1

Simplify $- (x + 3)$ .

Tap for more steps...

Step 2.4.1.1

Rewrite.

$x - 7 = 0 + 0 - (x + 3)$

Step 2.4.1.2

Simplify by adding zeros.

$x - 7 = - (x + 3)$

Step 2.4.1.3

Apply the distributive property.

$x - 7 = - x - 1 \cdot 3$

Step 2.4.1.4

Multiply $- 1$ by $3$ .

$x - 7 = - x - 3$

Step 2.4.2

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

Tap for more steps...

Step 2.4.2.1

Add $x$ to both sides of the equation.

$x - 7 + x = - 3$

Step 2.4.2.2

Add $x$ and $x$ .

$2 x - 7 = - 3$

Step 2.4.3

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

Tap for more steps...

Step 2.4.3.1

Add $7$ to both sides of the equation.

$2 x = - 3 + 7$

Step 2.4.3.2

Add $- 3$ and $7$ .

$2 x = 4$

Step 2.4.4

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

Tap for more steps...

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.

Tap for more steps...

Step 2.4.4.2.1

Cancel the common factor of $2$ .

Tap for more steps...

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.

Tap for more steps...

Step 2.4.4.3.1

Divide $4$ by $2$ .

$x = 2$

Step 2.5

List all of the solutions.

$x = 2$