Solve by Factoring square root of x+16=x-4

$\sqrt{x + 16} = x - 4$

Step 1

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

Subtract $x$ from both sides of the equation.

$\sqrt{x + 16} - x = - 4$

Add $4$ to both sides of the equation.

$\sqrt{x + 16} - x + 4 = 0$

Step 2

Move all terms not containing $\sqrt{x + 16}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{x + 16} + 4 = x$

Subtract $4$ from both sides of the equation.

$\sqrt{x + 16} = x - 4$

Step 3

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

${\sqrt{x + 16}}^{2} = {(x - 4)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 16}$ as ${(x + 16)}^{\frac{1}{2}}$ .

${({(x + 16)}^{\frac{1}{2}})}^{2} = {(x - 4)}^{2}$

Simplify the left side.

Simplify ${({(x + 16)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 16)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 16)}^{\frac{1}{2} \cdot 2} = {(x - 4)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 16)}^{\frac{1}{2} \cdot 2} = {(x - 4)}^{2}$

Rewrite the expression.

${(x + 16)}^{1} = {(x - 4)}^{2}$

Simplify.

$x + 16 = {(x - 4)}^{2}$

Simplify the right side.

Simplify ${(x - 4)}^{2}$ .

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$x + 16 = (x - 4) (x - 4)$

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$x + 16 = x (x - 4) - 4 (x - 4)$

Apply the distributive property.

$x + 16 = x \cdot x + x \cdot - 4 - 4 (x - 4)$

Apply the distributive property.

$x + 16 = x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x + 16 = x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4$

Move $- 4$ to the left of $x$ .

$x + 16 = x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$x + 16 = x^{2} - 4 x - 4 x + 16$

Subtract $4 x$ from $- 4 x$ .

$x + 16 = x^{2} - 8 x + 16$

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.

$x^{2} - 8 x + 16 = x + 16$

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

Subtract $x$ from both sides of the equation.

$x^{2} - 8 x + 16 - x = 16$

Subtract $x$ from $- 8 x$ .

$x^{2} - 9 x + 16 = 16$

Subtract $16$ from both sides of the equation.

$x^{2} - 9 x + 16 - 16 = 0$

Combine the opposite terms in $x^{2} - 9 x + 16 - 16$ .

Subtract $16$ from $16$ .

$x^{2} - 9 x + 0 = 0$

Add $x^{2} - 9 x$ and $0$ .

$x^{2} - 9 x = 0$

Factor $x$ out of $x^{2} - 9 x$ .

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

$x \cdot x - 9 x = 0$

Factor $x$ out of $- 9 x$ .

$x \cdot x + x \cdot - 9 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 9$ .

$x (x - 9) = 0$

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

$x = 0$

$x - 9 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

The final solution is all the values that make $x (x - 9) = 0$ true.

$x = 0, 9$

Step 6

Exclude the solutions that do not make $\sqrt{x + 16} - x + 4 = 0$ true.

$x = 9$

Solve by Factoring square root of x+16=x-4

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