Solve by Factoring square root of x-11+ square root of x+4=5

$\sqrt{x - 11} + \sqrt{x + 4} = 5$

Step 1

Subtract $5$ from both sides of the equation.

$\sqrt{x - 11} + \sqrt{x + 4} - 5 = 0$

Step 2

Move all terms not containing $\sqrt{x - 11}$ to the right side of the equation.

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

$\sqrt{x - 11} - 5 = - \sqrt{x + 4}$

Add $5$ to both sides of the equation.

$\sqrt{x - 11} = - \sqrt{x + 4} + 5$

Step 3

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

${\sqrt{x - 11}}^{2} = {(- \sqrt{x + 4} + 5)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 11}$ as ${(x - 11)}^{\frac{1}{2}}$ .

${({(x - 11)}^{\frac{1}{2}})}^{2} = {(- \sqrt{x + 4} + 5)}^{2}$

Simplify the left side.

Simplify ${({(x - 11)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x - 11)}^{\frac{1}{2}})}^{2}$ .

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

${(x - 11)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{x + 4} + 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 11)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{x + 4} + 5)}^{2}$

Rewrite the expression.

${(x - 11)}^{1} = {(- \sqrt{x + 4} + 5)}^{2}$

Simplify.

$x - 11 = {(- \sqrt{x + 4} + 5)}^{2}$

Simplify the right side.

Simplify ${(- \sqrt{x + 4} + 5)}^{2}$ .

Rewrite ${(- \sqrt{x + 4} + 5)}^{2}$ as $(- \sqrt{x + 4} + 5) (- \sqrt{x + 4} + 5)$ .

$x - 11 = (- \sqrt{x + 4} + 5) (- \sqrt{x + 4} + 5)$

Expand $(- \sqrt{x + 4} + 5) (- \sqrt{x + 4} + 5)$ using the FOIL Method.

Apply the distributive property.

$x - 11 = - \sqrt{x + 4} (- \sqrt{x + 4} + 5) + 5 (- \sqrt{x + 4} + 5)$

Apply the distributive property.

$x - 11 = - \sqrt{x + 4} (- \sqrt{x + 4}) - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4} + 5)$

Apply the distributive property.

$x - 11 = - \sqrt{x + 4} (- \sqrt{x + 4}) - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

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

Multiply $- 1$ by $- 1$ .

$x - 11 = 1 \sqrt{x + 4} \sqrt{x + 4} - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

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

$x - 11 = \sqrt{x + 4} \sqrt{x + 4} - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

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

$x - 11 = {\sqrt{x + 4}}^{1} \sqrt{x + 4} - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

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

$x - 11 = {\sqrt{x + 4}}^{1} {\sqrt{x + 4}}^{1} - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

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

$x - 11 = {\sqrt{x + 4}}^{1 + 1} - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

Add $1$ and $1$ .

$x - 11 = {\sqrt{x + 4}}^{2} - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

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

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

$x - 11 = {({(x + 4)}^{\frac{1}{2}})}^{2} - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

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

$x - 11 = {(x + 4)}^{\frac{1}{2} \cdot 2} - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

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

$x - 11 = {(x + 4)}^{\frac{2}{2}} - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - 11 = {(x + 4)}^{\frac{2}{2}} - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

Rewrite the expression.

$x - 11 = {(x + 4)}^{1} - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

Simplify.

$x - 11 = x + 4 - \sqrt{x + 4} \cdot 5 + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

Multiply $5$ by $- 1$ .

$x - 11 = x + 4 - 5 \sqrt{x + 4} + 5 (- \sqrt{x + 4}) + 5 \cdot 5$

Multiply $- 1$ by $5$ .

$x - 11 = x + 4 - 5 \sqrt{x + 4} - 5 \sqrt{x + 4} + 5 \cdot 5$

Multiply $5$ by $5$ .

$x - 11 = x + 4 - 5 \sqrt{x + 4} - 5 \sqrt{x + 4} + 25$

Add $4$ and $25$ .

$x - 11 = x + 29 - 5 \sqrt{x + 4} - 5 \sqrt{x + 4}$

Subtract $5 \sqrt{x + 4}$ from $- 5 \sqrt{x + 4}$ .

$x - 11 = x + 29 - 10 \sqrt{x + 4}$

Step 5

Solve for $- 10 \sqrt{x + 4}$ .

Rewrite the equation as $x + 29 - 10 \sqrt{x + 4} = x - 11$ .

$x + 29 - 10 \sqrt{x + 4} = x - 11$

Move all terms not containing $- 10 \sqrt{x + 4}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$29 - 10 \sqrt{x + 4} = x - 11 - x$

Subtract $29$ from both sides of the equation.

$- 10 \sqrt{x + 4} = x - 11 - x - 29$

Combine the opposite terms in $x - 11 - x - 29$ .

Subtract $x$ from $x$ .

$- 10 \sqrt{x + 4} = 0 - 11 - 29$

Subtract $11$ from $0$ .

$- 10 \sqrt{x + 4} = - 11 - 29$

Subtract $29$ from $- 11$ .

$- 10 \sqrt{x + 4} = - 40$

Step 6

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

${(- 10 \sqrt{x + 4})}^{2} = {(- 40)}^{2}$

Step 7

Simplify each side of the equation.

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

${(- 10 {(x + 4)}^{\frac{1}{2}})}^{2} = {(- 40)}^{2}$

Simplify the left side.

Simplify ${(- 10 {(x + 4)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 10 {(x + 4)}^{\frac{1}{2}}$ .

${(- 10)}^{2} {({(x + 4)}^{\frac{1}{2}})}^{2} = {(- 40)}^{2}$

Raise $- 10$ to the power of $2$ .

$100 {({(x + 4)}^{\frac{1}{2}})}^{2} = {(- 40)}^{2}$

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

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

$100 {(x + 4)}^{\frac{1}{2} \cdot 2} = {(- 40)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$100 {(x + 4)}^{\frac{1}{2} \cdot 2} = {(- 40)}^{2}$

Rewrite the expression.

$100 {(x + 4)}^{1} = {(- 40)}^{2}$

Simplify.

$100 (x + 4) = {(- 40)}^{2}$

Apply the distributive property.

$100 x + 100 \cdot 4 = {(- 40)}^{2}$

Multiply $100$ by $4$ .

$100 x + 400 = {(- 40)}^{2}$

Simplify the right side.

Raise $- 40$ to the power of $2$ .

$100 x + 400 = 1600$

Step 8

Solve for $x$ .

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

Subtract $400$ from both sides of the equation.

$100 x = 1600 - 400$

Subtract $400$ from $1600$ .

$100 x = 1200$

Divide each term in $100 x = 1200$ by $100$ and simplify.

Divide each term in $100 x = 1200$ by $100$ .

$\frac{100 x}{100} = \frac{1200}{100}$

Simplify the left side.

Cancel the common factor of $100$ .

Cancel the common factor.

$\frac{100 x}{100} = \frac{1200}{100}$

Divide $x$ by $1$ .

$x = \frac{1200}{100}$

Simplify the right side.

Divide $1200$ by $100$ .

$x = 12$

Solve by Factoring square root of x-11+ square root of x+4=5

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