Solve by Factoring x = square root of 2x+24

$x = \sqrt{2 x + 24}$

Step 1

Subtract $\sqrt{2 x + 24}$ from both sides of the equation.

$x - \sqrt{2 x + 24} = 0$

Step 2

Factor $2$ out of $2 x + 24$ .

Factor $2$ out of $2 x$ .

$x - \sqrt{2 (x) + 24} = 0$

Factor $2$ out of $24$ .

$x - \sqrt{2 x + 2 \cdot 12} = 0$

Factor $2$ out of $2 x + 2 \cdot 12$ .

$x - \sqrt{2 (x + 12)} = 0$

Step 3

Subtract $x$ from both sides of the equation.

$- \sqrt{2 (x + 12)} = - x$

Step 4

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

${(- \sqrt{2 (x + 12)})}^{2} = {(- x)}^{2}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (x + 12)}$ as ${(2 (x + 12))}^{\frac{1}{2}}$ .

${(- {(2 (x + 12))}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Simplify the left side.

Simplify ${(- {(2 (x + 12))}^{\frac{1}{2}})}^{2}$ .

Apply the distributive property.

${(- {(2 x + 2 \cdot 12)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Simplify the expression.

Multiply $2$ by $12$ .

${(- {(2 x + 24)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Apply the product rule to $- {(2 x + 24)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(2 x + 24)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {({(2 x + 24)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Multiply ${({(2 x + 24)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(2 x + 24)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Multiply the exponents in ${({(2 x + 24)}^{\frac{1}{2}})}^{2}$ .

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

${(2 x + 24)}^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x + 24)}^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Rewrite the expression.

${(2 x + 24)}^{1} = {(- x)}^{2}$

Simplify.

$2 x + 24 = {(- x)}^{2}$

Simplify the right side.

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

Apply the product rule to $- x$ .

$2 x + 24 = {(- 1)}^{2} x^{2}$

Raise $- 1$ to the power of $2$ .

$2 x + 24 = 1 x^{2}$

Multiply $x^{2}$ by $1$ .

$2 x + 24 = x^{2}$

Step 6

Solve for $x$ .

Subtract $x^{2}$ from both sides of the equation.

$2 x + 24 - x^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $2 x + 24 - x^{2}$ .

Reorder the expression.

Move $24$ .

$2 x - x^{2} + 24 = 0$

Reorder $2 x$ and $- x^{2}$ .

$- x^{2} + 2 x + 24 = 0$

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 2 x + 24 = 0$

Factor $- 1$ out of $2 x$ .

$- (x^{2}) - (- 2 x) + 24 = 0$

Rewrite $24$ as $- 1 (- 24)$ .

$- (x^{2}) - (- 2 x) - 1 \cdot - 24 = 0$

Factor $- 1$ out of $- (x^{2}) - (- 2 x)$ .

$- (x^{2} - 2 x) - 1 \cdot - 24 = 0$

Factor $- 1$ out of $- (x^{2} - 2 x) - 1 (- 24)$ .

$- (x^{2} - 2 x - 24) = 0$

Factor.

Factor $x^{2} - 2 x - 24$ 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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Write the factored form using these integers.

$- ((x - 6) (x + 4)) = 0$

Remove unnecessary parentheses.

$- (x - 6) (x + 4) = 0$

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

$x - 6 = 0$

$x + 4 = 0$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $- (x - 6) (x + 4) = 0$ true.

$x = 6, - 4$

Step 7

Exclude the solutions that do not make $x - \sqrt{2 (x + 12)} = 0$ true.

$x = 6$

Solve by Factoring x = square root of 2x+24

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