Solve for x -12=-x+ square root of 2x+24

$- 12 = - x + \sqrt{2 x + 24}$

Step 1

Rewrite the equation as $- x + \sqrt{2 x + 24} = - 12$ .

$- x + \sqrt{2 x + 24} = - 12$

Step 2

Add $x$ to both sides of the equation.

$\sqrt{2 x + 24} = - 12 + x$

Step 3

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

${\sqrt{2 x + 24}}^{2} = {(- 12 + x)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 24}$ as ${(2 x + 24)}^{\frac{1}{2}}$ .

${({(2 x + 24)}^{\frac{1}{2}})}^{2} = {(- 12 + x)}^{2}$

Simplify the left side.

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

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

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

${(2 x + 24)}^{\frac{1}{2} \cdot 2} = {(- 12 + x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x + 24)}^{\frac{1}{2} \cdot 2} = {(- 12 + x)}^{2}$

Rewrite the expression.

${(2 x + 24)}^{1} = {(- 12 + x)}^{2}$

Simplify.

$2 x + 24 = {(- 12 + x)}^{2}$

Simplify the right side.

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

Rewrite ${(- 12 + x)}^{2}$ as $(- 12 + x) (- 12 + x)$ .

$2 x + 24 = (- 12 + x) (- 12 + x)$

Expand $(- 12 + x) (- 12 + x)$ using the FOIL Method.

Apply the distributive property.

$2 x + 24 = - 12 (- 12 + x) + x (- 12 + x)$

Apply the distributive property.

$2 x + 24 = - 12 \cdot - 12 - 12 x + x (- 12 + x)$

Apply the distributive property.

$2 x + 24 = - 12 \cdot - 12 - 12 x + x \cdot - 12 + x \cdot x$

Simplify and combine like terms.

Simplify each term.

Multiply $- 12$ by $- 12$ .

$2 x + 24 = 144 - 12 x + x \cdot - 12 + x \cdot x$

Move $- 12$ to the left of $x$ .

$2 x + 24 = 144 - 12 x - 12 \cdot x + x \cdot x$

Multiply $x$ by $x$ .

$2 x + 24 = 144 - 12 x - 12 x + x^{2}$

Subtract $12 x$ from $- 12 x$ .

$2 x + 24 = 144 - 24 x + x^{2}$

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.

$144 - 24 x + x^{2} = 2 x + 24$

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

Subtract $2 x$ from both sides of the equation.

$144 - 24 x + x^{2} - 2 x = 24$

Subtract $2 x$ from $- 24 x$ .

$144 + x^{2} - 26 x = 24$

Subtract $24$ from both sides of the equation.

$144 + x^{2} - 26 x - 24 = 0$

Subtract $24$ from $144$ .

$x^{2} - 26 x + 120 = 0$

Factor $x^{2} - 26 x + 120$ 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 $120$ and whose sum is $- 26$ .

$- 20, - 6$

Write the factored form using these integers.

$(x - 20) (x - 6) = 0$

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

$x - 20 = 0$

$x - 6 = 0$

Set $x - 20$ equal to $0$ and solve for $x$ .

Set $x - 20$ equal to $0$ .

$x - 20 = 0$

Add $20$ to both sides of the equation.

$x = 20$

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.

$x = 6$

The final solution is all the values that make $(x - 20) (x - 6) = 0$ true.

$x = 20, 6$

Step 6

Exclude the solutions that do not make $- 12 = - x + \sqrt{2 x + 24}$ true.

$x = 20$

Solve for x -12=-x+ square root of 2x+24

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