Solve by Factoring square root of x+8=x-4

$\sqrt{x + 8} = x - 4$

Step 1

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

Subtract $x$ from both sides of the equation.

$\sqrt{x + 8} - x = - 4$

Add $4$ to both sides of the equation.

$\sqrt{x + 8} - x + 4 = 0$

Step 2

Move all terms not containing $\sqrt{x + 8}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{x + 8} + 4 = x$

Subtract $4$ from both sides of the equation.

$\sqrt{x + 8} = x - 4$

Step 3

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

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

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 8}$ as ${(x + 8)}^{\frac{1}{2}}$ .

${({(x + 8)}^{\frac{1}{2}})}^{2} = {(x - 4)}^{2}$

Simplify the left side.

Simplify ${({(x + 8)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 8)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 8)}^{\frac{1}{2} \cdot 2} = {(x - 4)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 8)}^{\frac{1}{2} \cdot 2} = {(x - 4)}^{2}$

Rewrite the expression.

${(x + 8)}^{1} = {(x - 4)}^{2}$

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x + 8 = x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4$

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

$x + 8 = x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$x + 8 = x^{2} - 4 x - 4 x + 16$

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

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

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.

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

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

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $- 8 x$ .

$x^{2} - 9 x + 16 = 8$

Subtract $8$ from both sides of the equation.

$x^{2} - 9 x + 16 - 8 = 0$

Subtract $8$ from $16$ .

$x^{2} - 9 x + 8 = 0$

Factor $x^{2} - 9 x + 8$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $8$ and whose sum is $- 9$ .

$- 8, - 1$

Write the factored form using these integers.

$(x - 8) (x - 1) = 0$

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

$x - 8 = 0$

$x - 1 = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $(x - 8) (x - 1) = 0$ true.

$x = 8, 1$

Step 6

Exclude the solutions that do not make $\sqrt{x + 8} - x + 4 = 0$ true.

$x = 8$

Solve by Factoring square root of x+8=x-4

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