Solve by Factoring square root of x+64=x-8

$\sqrt{x + 64} = x - 8$

Step 1

Move all the expressions to the left side of the equation.

Subtract $x$ from both sides of the equation.

$\sqrt{x + 64} - x = - 8$

Add $8$ to both sides of the equation.

$\sqrt{x + 64} - x + 8 = 0$

Step 2

Move all terms not containing $\sqrt{x + 64}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{x + 64} + 8 = x$

Subtract $8$ from both sides of the equation.

$\sqrt{x + 64} = x - 8$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x + 64}}^{2} = {(x - 8)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 64}$ as ${(x + 64)}^{\frac{1}{2}}$ .

${({(x + 64)}^{\frac{1}{2}})}^{2} = {(x - 8)}^{2}$

Simplify the left side.

Simplify ${({(x + 64)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 64)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x + 64)}^{\frac{1}{2} \cdot 2} = {(x - 8)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 64)}^{\frac{1}{2} \cdot 2} = {(x - 8)}^{2}$

Rewrite the expression.

${(x + 64)}^{1} = {(x - 8)}^{2}$

Simplify.

$x + 64 = {(x - 8)}^{2}$

Simplify the right side.

Simplify ${(x - 8)}^{2}$ .

Rewrite ${(x - 8)}^{2}$ as $(x - 8) (x - 8)$ .

$x + 64 = (x - 8) (x - 8)$

Expand $(x - 8) (x - 8)$ using the FOIL Method.

Apply the distributive property.

$x + 64 = x (x - 8) - 8 (x - 8)$

Apply the distributive property.

$x + 64 = x \cdot x + x \cdot - 8 - 8 (x - 8)$

Apply the distributive property.

$x + 64 = x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x + 64 = x^{2} + x \cdot - 8 - 8 x - 8 \cdot - 8$

Move $- 8$ to the left of $x$ .

$x + 64 = x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8$

Multiply $- 8$ by $- 8$ .

$x + 64 = x^{2} - 8 x - 8 x + 64$

Subtract $8 x$ from $- 8 x$ .

$x + 64 = x^{2} - 16 x + 64$

Step 5

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 16 x + 64 = x + 64$

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

Subtract $x$ from both sides of the equation.

$x^{2} - 16 x + 64 - x = 64$

Subtract $x$ from $- 16 x$ .

$x^{2} - 17 x + 64 = 64$

Subtract $64$ from both sides of the equation.

$x^{2} - 17 x + 64 - 64 = 0$

Combine the opposite terms in $x^{2} - 17 x + 64 - 64$ .

Subtract $64$ from $64$ .

$x^{2} - 17 x + 0 = 0$

Add $x^{2} - 17 x$ and $0$ .

$x^{2} - 17 x = 0$

Factor $x$ out of $x^{2} - 17 x$ .

Factor $x$ out of $x^{2}$ .

$x \cdot x - 17 x = 0$

Factor $x$ out of $- 17 x$ .

$x \cdot x + x \cdot - 17 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 17$ .

$x (x - 17) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 17 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 17$ equal to $0$ and solve for $x$ .

Set $x - 17$ equal to $0$ .

$x - 17 = 0$

Add $17$ to both sides of the equation.

$x = 17$

The final solution is all the values that make $x (x - 17) = 0$ true.

$x = 0, 17$

Step 6

Exclude the solutions that do not make $\sqrt{x + 64} - x + 8 = 0$ true.

$x = 17$

Solve by Factoring square root of x+64=x-8

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