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

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

Step 1

Subtract $4$ from both sides of the equation.

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

Step 2

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

Subtract $\sqrt{x - 4}$ from both sides of the equation.

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

Add $4$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${({(x + 4)}^{\frac{1}{2}})}^{2}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(- \sqrt{x - 4} + 4)}^{2}$ .

Rewrite ${(- \sqrt{x - 4} + 4)}^{2}$ as $(- \sqrt{x - 4} + 4) (- \sqrt{x - 4} + 4)$ .

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

Expand $(- \sqrt{x - 4} + 4) (- \sqrt{x - 4} + 4)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- \sqrt{x - 4} (- \sqrt{x - 4})$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\sqrt{x - 4}$ by $1$ .

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

Raise $\sqrt{x - 4}$ to the power of $1$ .

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

Raise $\sqrt{x - 4}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{x - 4}}^{2}$ as $x - 4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Multiply $4$ by $- 1$ .

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

Multiply $- 1$ by $4$ .

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

Multiply $4$ by $4$ .

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

Add $- 4$ and $16$ .

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

Subtract $4 \sqrt{x - 4}$ from $- 4 \sqrt{x - 4}$ .

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

Step 5

Solve for $- 8 \sqrt{x - 4}$ .

Rewrite the equation as $x + 12 - 8 \sqrt{x - 4} = x + 4$ .

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

Move all terms not containing $- 8 \sqrt{x - 4}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

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

Subtract $12$ from both sides of the equation.

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

Combine the opposite terms in $x + 4 - x - 12$ .

Subtract $x$ from $x$ .

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

Add $0$ and $4$ .

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

Subtract $12$ from $4$ .

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

Step 6

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

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

Step 7

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 4}$ as ${(x - 4)}^{\frac{1}{2}}$ .

${(- 8 {(x - 4)}^{\frac{1}{2}})}^{2} = {(- 8)}^{2}$

Simplify the left side.

Simplify ${(- 8 {(x - 4)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 8 {(x - 4)}^{\frac{1}{2}}$ .

${(- 8)}^{2} {({(x - 4)}^{\frac{1}{2}})}^{2} = {(- 8)}^{2}$

Raise $- 8$ to the power of $2$ .

$64 {({(x - 4)}^{\frac{1}{2}})}^{2} = {(- 8)}^{2}$

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

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

$64 {(x - 4)}^{\frac{1}{2} \cdot 2} = {(- 8)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$64 {(x - 4)}^{\frac{1}{2} \cdot 2} = {(- 8)}^{2}$

Rewrite the expression.

$64 {(x - 4)}^{1} = {(- 8)}^{2}$

Simplify.

$64 (x - 4) = {(- 8)}^{2}$

Apply the distributive property.

$64 x + 64 \cdot - 4 = {(- 8)}^{2}$

Multiply $64$ by $- 4$ .

$64 x - 256 = {(- 8)}^{2}$

Simplify the right side.

Raise $- 8$ to the power of $2$ .

$64 x - 256 = 64$

Step 8

Solve for $x$ .

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

Add $256$ to both sides of the equation.

$64 x = 64 + 256$

Add $64$ and $256$ .

$64 x = 320$

Divide each term in $64 x = 320$ by $64$ and simplify.

Divide each term in $64 x = 320$ by $64$ .

$\frac{64 x}{64} = \frac{320}{64}$

Simplify the left side.

Cancel the common factor of $64$ .

Cancel the common factor.

$\frac{64 x}{64} = \frac{320}{64}$

Divide $x$ by $1$ .

$x = \frac{320}{64}$

Simplify the right side.

Divide $320$ by $64$ .

$x = 5$

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

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