Solve for x square root of x+14=x-6

$\sqrt{x + 14} = x - 6$

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

${\sqrt{x + 14}}^{2} = {(x - 6)}^{2}$

Simplify each side of the equation.

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Multiply the exponents in ${({(x + 14)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 14)}^{\frac{1}{2} \cdot 2} = {(x - 6)}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

${(x + 14)}^{\frac{1}{2} \cdot 2} = {(x - 6)}^{2}$

Rewrite the expression.

${(x + 14)}^{1} = {(x - 6)}^{2}$

Simplify.

$x + 14 = {(x - 6)}^{2}$

Solve for $x$ .

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Simplify ${(x - 6)}^{2}$ .

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Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$x + 14 = (x - 6) (x - 6)$

Expand $(x - 6) (x - 6)$ using the FOIL Method.

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Apply the distributive property.

$x + 14 = x (x - 6) - 6 (x - 6)$

Apply the distributive property.

$x + 14 = x \cdot x + x \cdot - 6 - 6 (x - 6)$

Apply the distributive property.

$x + 14 = x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$x + 14 = x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6$

Move $- 6$ to the left of $x$ .

$x + 14 = x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$x + 14 = x^{2} - 6 x - 6 x + 36$

Subtract $6 x$ from $- 6 x$ .

$x + 14 = x^{2} - 12 x + 36$

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

$x^{2} - 12 x + 36 = x + 14$

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

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Subtract $x$ from both sides of the equation.

$x^{2} - 12 x + 36 - x = 14$

Subtract $x$ from $- 12 x$ .

$x^{2} - 13 x + 36 = 14$

Move $14$ to the left side of the equation by subtracting it from both sides.

$x^{2} - 13 x + 36 - 14 = 0$

Subtract $14$ from $36$ .

$x^{2} - 13 x + 22 = 0$

Factor $x^{2} - 13 x + 22$ using the AC method.

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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 $22$ and whose sum is $- 13$ .

$- 11, - 2$

Write the factored form using these integers.

$(x - 11) (x - 2) = 0$

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

$x - 11 = 0$

$x - 2 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$x - 11 = 0$

Add $11$ to both sides of the equation.

$x = 11$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(x - 11) (x - 2) = 0$ true.

$x = 11, 2$

Exclude the solutions that do not make $\sqrt{x + 14} = x - 6$ true.

$x = 11$

Solve for x square root of x+14=x-6

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