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

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

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

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

Move all terms not containing $\sqrt{x - 2}$ to the right side of the equation.

Subtract $12$ from both sides of the equation.

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

Subtract $12$ from $- 2$ .

$\sqrt{x - 2} = x - 14$

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

${\sqrt{x - 2}}^{2} = {(x - 14)}^{2}$

Simplify each side of the equation.

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

${({(x - 2)}^{\frac{1}{2}})}^{2} = {(x - 14)}^{2}$

Simplify the left side.

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

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

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

${(x - 2)}^{\frac{1}{2} \cdot 2} = {(x - 14)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 2)}^{\frac{1}{2} \cdot 2} = {(x - 14)}^{2}$

Rewrite the expression.

${(x - 2)}^{1} = {(x - 14)}^{2}$

Simplify.

$x - 2 = {(x - 14)}^{2}$

Simplify the right side.

Simplify ${(x - 14)}^{2}$ .

Rewrite ${(x - 14)}^{2}$ as $(x - 14) (x - 14)$ .

$x - 2 = (x - 14) (x - 14)$

Expand $(x - 14) (x - 14)$ using the FOIL Method.

Apply the distributive property.

$x - 2 = x (x - 14) - 14 (x - 14)$

Apply the distributive property.

$x - 2 = x \cdot x + x \cdot - 14 - 14 (x - 14)$

Apply the distributive property.

$x - 2 = x \cdot x + x \cdot - 14 - 14 x - 14 \cdot - 14$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x - 2 = x^{2} + x \cdot - 14 - 14 x - 14 \cdot - 14$

Move $- 14$ to the left of $x$ .

$x - 2 = x^{2} - 14 \cdot x - 14 x - 14 \cdot - 14$

Multiply $- 14$ by $- 14$ .

$x - 2 = x^{2} - 14 x - 14 x + 196$

Subtract $14 x$ from $- 14 x$ .

$x - 2 = x^{2} - 28 x + 196$

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.

$x^{2} - 28 x + 196 = x - 2$

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

Subtract $x$ from both sides of the equation.

$x^{2} - 28 x + 196 - x = - 2$

Subtract $x$ from $- 28 x$ .

$x^{2} - 29 x + 196 = - 2$

Add $2$ to both sides of the equation.

$x^{2} - 29 x + 196 + 2 = 0$

Add $196$ and $2$ .

$x^{2} - 29 x + 198 = 0$

Factor $x^{2} - 29 x + 198$ 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 $198$ and whose sum is $- 29$ .

$- 18, - 11$

Write the factored form using these integers.

$(x - 18) (x - 11) = 0$

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

$x - 18 = 0$

$x - 11 = 0$

Set $x - 18$ equal to $0$ and solve for $x$ .

Set $x - 18$ equal to $0$ .

$x - 18 = 0$

Add $18$ to both sides of the equation.

$x = 18$

Set $x - 11$ equal to $0$ and solve for $x$ .

Set $x - 11$ equal to $0$ .

$x - 11 = 0$

Add $11$ to both sides of the equation.

$x = 11$

The final solution is all the values that make $(x - 18) (x - 11) = 0$ true.

$x = 18, 11$

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

$x = 18$

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

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