Solve by Factoring square root of x-9=5- square root of x+6

$\sqrt{x - 9} = 5 - \sqrt{x + 6}$

Step 1

Move all the expressions to the left side of the equation.

Subtract $5$ from both sides of the equation.

$\sqrt{x - 9} - 5 = - \sqrt{x + 6}$

Add $\sqrt{x + 6}$ to both sides of the equation.

$\sqrt{x - 9} - 5 + \sqrt{x + 6} = 0$

Step 2

Move all terms not containing $\sqrt{x - 9}$ to the right side of the equation.

Add $5$ to both sides of the equation.

$\sqrt{x - 9} + \sqrt{x + 6} = 5$

Subtract $\sqrt{x + 6}$ from both sides of the equation.

$\sqrt{x - 9} = 5 - \sqrt{x + 6}$

Step 3

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

${\sqrt{x - 9}}^{2} = {(5 - \sqrt{x + 6})}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 9}$ as ${(x - 9)}^{\frac{1}{2}}$ .

${({(x - 9)}^{\frac{1}{2}})}^{2} = {(5 - \sqrt{x + 6})}^{2}$

Simplify the left side.

Simplify ${({(x - 9)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x - 9)}^{\frac{1}{2}})}^{2}$ .

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

${(x - 9)}^{\frac{1}{2} \cdot 2} = {(5 - \sqrt{x + 6})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 9)}^{\frac{1}{2} \cdot 2} = {(5 - \sqrt{x + 6})}^{2}$

Rewrite the expression.

${(x - 9)}^{1} = {(5 - \sqrt{x + 6})}^{2}$

Simplify.

$x - 9 = {(5 - \sqrt{x + 6})}^{2}$

Simplify the right side.

Simplify ${(5 - \sqrt{x + 6})}^{2}$ .

Rewrite ${(5 - \sqrt{x + 6})}^{2}$ as $(5 - \sqrt{x + 6}) (5 - \sqrt{x + 6})$ .

$x - 9 = (5 - \sqrt{x + 6}) (5 - \sqrt{x + 6})$

Expand $(5 - \sqrt{x + 6}) (5 - \sqrt{x + 6})$ using the FOIL Method.

Apply the distributive property.

$x - 9 = 5 (5 - \sqrt{x + 6}) - \sqrt{x + 6} (5 - \sqrt{x + 6})$

Apply the distributive property.

$x - 9 = 5 \cdot 5 + 5 (- \sqrt{x + 6}) - \sqrt{x + 6} (5 - \sqrt{x + 6})$

Apply the distributive property.

$x - 9 = 5 \cdot 5 + 5 (- \sqrt{x + 6}) - \sqrt{x + 6} \cdot 5 - \sqrt{x + 6} (- \sqrt{x + 6})$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$x - 9 = 25 + 5 (- \sqrt{x + 6}) - \sqrt{x + 6} \cdot 5 - \sqrt{x + 6} (- \sqrt{x + 6})$

Multiply $- 1$ by $5$ .

$x - 9 = 25 - 5 \sqrt{x + 6} - \sqrt{x + 6} \cdot 5 - \sqrt{x + 6} (- \sqrt{x + 6})$

Multiply $5$ by $- 1$ .

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} - \sqrt{x + 6} (- \sqrt{x + 6})$

Multiply $- \sqrt{x + 6} (- \sqrt{x + 6})$ .

Multiply $- 1$ by $- 1$ .

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + 1 \sqrt{x + 6} \sqrt{x + 6}$

Multiply $\sqrt{x + 6}$ by $1$ .

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + \sqrt{x + 6} \sqrt{x + 6}$

Raise $\sqrt{x + 6}$ to the power of $1$ .

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + {\sqrt{x + 6}}^{1} \sqrt{x + 6}$

Raise $\sqrt{x + 6}$ to the power of $1$ .

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + {\sqrt{x + 6}}^{1} {\sqrt{x + 6}}^{1}$

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

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + {\sqrt{x + 6}}^{1 + 1}$

Add $1$ and $1$ .

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + {\sqrt{x + 6}}^{2}$

Rewrite ${\sqrt{x + 6}}^{2}$ as $x + 6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 6}$ as ${(x + 6)}^{\frac{1}{2}}$ .

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + {({(x + 6)}^{\frac{1}{2}})}^{2}$

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

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + {(x + 6)}^{\frac{1}{2} \cdot 2}$

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

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + {(x + 6)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + {(x + 6)}^{\frac{2}{2}}$

Rewrite the expression.

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + {(x + 6)}^{1}$

Simplify.

$x - 9 = 25 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + x + 6$

Add $25$ and $6$ .

$x - 9 = 31 - 5 \sqrt{x + 6} - 5 \sqrt{x + 6} + x$

Subtract $5 \sqrt{x + 6}$ from $- 5 \sqrt{x + 6}$ .

$x - 9 = 31 - 10 \sqrt{x + 6} + x$

Step 5

Solve for $- 10 \sqrt{x + 6}$ .

Rewrite the equation as $31 - 10 \sqrt{x + 6} + x = x - 9$ .

$31 - 10 \sqrt{x + 6} + x = x - 9$

Move all terms not containing $- 10 \sqrt{x + 6}$ to the right side of the equation.

Subtract $31$ from both sides of the equation.

$- 10 \sqrt{x + 6} + x = x - 9 - 31$

Subtract $x$ from both sides of the equation.

$- 10 \sqrt{x + 6} = x - 9 - 31 - x$

Combine the opposite terms in $x - 9 - 31 - x$ .

Subtract $x$ from $x$ .

$- 10 \sqrt{x + 6} = 0 - 9 - 31$

Subtract $9$ from $0$ .

$- 10 \sqrt{x + 6} = - 9 - 31$

Subtract $31$ from $- 9$ .

$- 10 \sqrt{x + 6} = - 40$

Step 6

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

${(- 10 \sqrt{x + 6})}^{2} = {(- 40)}^{2}$

Step 7

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 6}$ as ${(x + 6)}^{\frac{1}{2}}$ .

${(- 10 {(x + 6)}^{\frac{1}{2}})}^{2} = {(- 40)}^{2}$

Simplify the left side.

Simplify ${(- 10 {(x + 6)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 10 {(x + 6)}^{\frac{1}{2}}$ .

${(- 10)}^{2} {({(x + 6)}^{\frac{1}{2}})}^{2} = {(- 40)}^{2}$

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

$100 {({(x + 6)}^{\frac{1}{2}})}^{2} = {(- 40)}^{2}$

Multiply the exponents in ${({(x + 6)}^{\frac{1}{2}})}^{2}$ .

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

$100 {(x + 6)}^{\frac{1}{2} \cdot 2} = {(- 40)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$100 {(x + 6)}^{\frac{1}{2} \cdot 2} = {(- 40)}^{2}$

Rewrite the expression.

$100 {(x + 6)}^{1} = {(- 40)}^{2}$

Simplify.

$100 (x + 6) = {(- 40)}^{2}$

Apply the distributive property.

$100 x + 100 \cdot 6 = {(- 40)}^{2}$

Multiply $100$ by $6$ .

$100 x + 600 = {(- 40)}^{2}$

Simplify the right side.

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

$100 x + 600 = 1600$

Step 8

Solve for $x$ .

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

Subtract $600$ from both sides of the equation.

$100 x = 1600 - 600$

Subtract $600$ from $1600$ .

$100 x = 1000$

Divide each term in $100 x = 1000$ by $100$ and simplify.

Divide each term in $100 x = 1000$ by $100$ .

$\frac{100 x}{100} = \frac{1000}{100}$

Simplify the left side.

Cancel the common factor of $100$ .

Cancel the common factor.

$\frac{100 x}{100} = \frac{1000}{100}$

Divide $x$ by $1$ .

$x = \frac{1000}{100}$

Simplify the right side.

Divide $1000$ by $100$ .

$x = 10$

Solve by Factoring square root of x-9=5- square root of x+6

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