Solve by Factoring square root of x+4+ square root of x-12=8

$\sqrt{x + 4} + \sqrt{x - 12} = 8$

Move $8$ to the left side of the equation by subtracting it from both sides.

$\sqrt{x + 4} + \sqrt{x - 12} - 8 = 0$

Move all terms not containing $\sqrt{x + 4}$ to the right side of the equation.

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Subtract $\sqrt{x - 12}$ from both sides of the equation.

$\sqrt{x + 4} - 8 = - \sqrt{x - 12}$

Add $8$ to both sides of the equation.

$\sqrt{x + 4} = - \sqrt{x - 12} + 8$

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

${\sqrt{x + 4}}^{2} = {(- \sqrt{x - 12} + 8)}^{2}$

Simplify each side of the equation.

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Multiply the exponents in ${({(x + 4)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 4)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{x - 12} + 8)}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

${(x + 4)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{x - 12} + 8)}^{2}$

Rewrite the expression.

${(x + 4)}^{1} = {(- \sqrt{x - 12} + 8)}^{2}$

Simplify.

$x + 4 = {(- \sqrt{x - 12} + 8)}^{2}$

Solve for $x$ .

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Simplify ${(- \sqrt{x - 12} + 8)}^{2}$ .

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Rewrite ${(- \sqrt{x - 12} + 8)}^{2}$ as $(- \sqrt{x - 12} + 8) (- \sqrt{x - 12} + 8)$ .

$x + 4 = (- \sqrt{x - 12} + 8) (- \sqrt{x - 12} + 8)$

Expand $(- \sqrt{x - 12} + 8) (- \sqrt{x - 12} + 8)$ using the FOIL Method.

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Apply the distributive property.

$x + 4 = - \sqrt{x - 12} (- \sqrt{x - 12} + 8) + 8 (- \sqrt{x - 12} + 8)$

Apply the distributive property.

$x + 4 = - \sqrt{x - 12} (- \sqrt{x - 12}) - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12} + 8)$

Apply the distributive property.

$x + 4 = - \sqrt{x - 12} (- \sqrt{x - 12}) - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Simplify and combine like terms.

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Simplify each term.

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Multiply $- \sqrt{x - 12} (- \sqrt{x - 12})$ .

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Multiply $- 1$ by $- 1$ .

$x + 4 = 1 \sqrt{x - 12} \sqrt{x - 12} - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Multiply $\sqrt{x - 12}$ by $1$ .

$x + 4 = \sqrt{x - 12} \sqrt{x - 12} - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Raise $\sqrt{x - 12}$ to the power of $1$ .

$x + 4 = {\sqrt{x - 12}}^{1} \sqrt{x - 12} - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Raise $\sqrt{x - 12}$ to the power of $1$ .

$x + 4 = {\sqrt{x - 12}}^{1} {\sqrt{x - 12}}^{1} - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x + 4 = {\sqrt{x - 12}}^{1 + 1} - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Add $1$ and $1$ .

$x + 4 = {\sqrt{x - 12}}^{2} - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Rewrite ${\sqrt{x - 12}}^{2}$ as $x - 12$ .

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

$x + 4 = {({(x - 12)}^{\frac{1}{2}})}^{2} - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

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

$x + 4 = {(x - 12)}^{\frac{1}{2} \cdot 2} - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Combine $\frac{1}{2}$ and $2$ .

$x + 4 = {(x - 12)}^{\frac{2}{2}} - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Cancel the common factor of $2$ .

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Cancel the common factor.

$x + 4 = {(x - 12)}^{\frac{2}{2}} - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Divide $1$ by $1$ .

$x + 4 = {(x - 12)}^{1} - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Simplify.

$x + 4 = x - 12 - \sqrt{x - 12} \cdot 8 + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Multiply $8$ by $- 1$ .

$x + 4 = x - 12 - 8 \sqrt{x - 12} + 8 (- \sqrt{x - 12}) + 8 \cdot 8$

Multiply $- 1$ by $8$ .

$x + 4 = x - 12 - 8 \sqrt{x - 12} - 8 \sqrt{x - 12} + 8 \cdot 8$

Multiply $8$ by $8$ .

$x + 4 = x - 12 - 8 \sqrt{x - 12} - 8 \sqrt{x - 12} + 64$

Add $- 12$ and $64$ .

$x + 4 = x + 52 - 8 \sqrt{x - 12} - 8 \sqrt{x - 12}$

Subtract $8 \sqrt{x - 12}$ from $- 8 \sqrt{x - 12}$ .

$x + 4 = x + 52 - 16 \sqrt{x - 12}$

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

$x + 52 - 16 \sqrt{x - 12} = x + 4$

Move all terms not containing $\sqrt{x - 12}$ to the right side of the equation.

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Subtract $x$ from both sides of the equation.

$52 - 16 \sqrt{x - 12} = x + 4 - x$

Subtract $52$ from both sides of the equation.

$- 16 \sqrt{x - 12} = x + 4 - x - 52$

Combine the opposite terms in $x + 4 - x - 52$ .

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Subtract $x$ from $x$ .

$- 16 \sqrt{x - 12} = 0 + 4 - 52$

Add $0$ and $4$ .

$- 16 \sqrt{x - 12} = 4 - 52$

Subtract $52$ from $4$ .

$- 16 \sqrt{x - 12} = - 48$

Divide each term by $- 16$ and simplify.

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Divide each term in $- 16 \sqrt{x - 12} = - 48$ by $- 16$ .

$\frac{- 16 \sqrt{x - 12}}{- 16} = \frac{- 48}{- 16}$

Cancel the common factor of $- 16$ .

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Cancel the common factor.

$\frac{- 16 \sqrt{x - 12}}{- 16} = \frac{- 48}{- 16}$

Divide $\sqrt{x - 12}$ by $1$ .

$\sqrt{x - 12} = \frac{- 48}{- 16}$

Divide $- 48$ by $- 16$ .

$\sqrt{x - 12} = 3$

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

${\sqrt{x - 12}}^{2} = 3^{2}$

Simplify each side of the equation.

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Multiply the exponents in ${({(x - 12)}^{\frac{1}{2}})}^{2}$ .

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

${(x - 12)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

${(x - 12)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Rewrite the expression.

${(x - 12)}^{1} = 3^{2}$

Simplify.

$x - 12 = 3^{2}$

Raise $3$ to the power of $2$ .

$x - 12 = 9$

Move all terms not containing $x$ to the right side of the equation.

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Add $12$ to both sides of the equation.

$x = 9 + 12$

Add $9$ and $12$ .

$x = 21$

Solve by Factoring square root of x+4+ square root of x-12=8

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