Solve by Factoring 2 square root of x+8=3 square root of x-2

$2 \sqrt{x + 8} = 3 \sqrt{x - 2}$

Step 1

Subtract $3 \sqrt{x - 2}$ from both sides of the equation.

$2 \sqrt{x + 8} - 3 \sqrt{x - 2} = 0$

Step 2

Add $3 \sqrt{x - 2}$ to both sides of the equation.

$2 \sqrt{x + 8} = 3 \sqrt{x - 2}$

Step 3

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

${(2 \sqrt{x + 8})}^{2} = {(3 \sqrt{x - 2})}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 8}$ as ${(x + 8)}^{\frac{1}{2}}$ .

${(2 {(x + 8)}^{\frac{1}{2}})}^{2} = {(3 \sqrt{x - 2})}^{2}$

Simplify the left side.

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

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

$2^{2} {({(x + 8)}^{\frac{1}{2}})}^{2} = {(3 \sqrt{x - 2})}^{2}$

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

$4 {({(x + 8)}^{\frac{1}{2}})}^{2} = {(3 \sqrt{x - 2})}^{2}$

Multiply the exponents in ${({(x + 8)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(x + 8)}^{\frac{1}{2} \cdot 2} = {(3 \sqrt{x - 2})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(x + 8)}^{\frac{1}{2} \cdot 2} = {(3 \sqrt{x - 2})}^{2}$

Rewrite the expression.

$4 {(x + 8)}^{1} = {(3 \sqrt{x - 2})}^{2}$

Simplify.

$4 (x + 8) = {(3 \sqrt{x - 2})}^{2}$

Apply the distributive property.

$4 x + 4 \cdot 8 = {(3 \sqrt{x - 2})}^{2}$

Multiply $4$ by $8$ .

$4 x + 32 = {(3 \sqrt{x - 2})}^{2}$

Simplify the right side.

Simplify ${(3 \sqrt{x - 2})}^{2}$ .

Simplify the expression.

Apply the product rule to $3 \sqrt{x - 2}$ .

$4 x + 32 = 3^{2} {\sqrt{x - 2}}^{2}$

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

$4 x + 32 = 9 {\sqrt{x - 2}}^{2}$

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

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

$4 x + 32 = 9 {({(x - 2)}^{\frac{1}{2}})}^{2}$

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

$4 x + 32 = 9 {(x - 2)}^{\frac{1}{2} \cdot 2}$

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

$4 x + 32 = 9 {(x - 2)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 x + 32 = 9 {(x - 2)}^{\frac{2}{2}}$

Rewrite the expression.

$4 x + 32 = 9 {(x - 2)}^{1}$

Simplify.

$4 x + 32 = 9 (x - 2)$

Apply the distributive property.

$4 x + 32 = 9 x + 9 \cdot - 2$

Multiply $9$ by $- 2$ .

$4 x + 32 = 9 x - 18$

Step 5

Solve for $x$ .

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

Subtract $9 x$ from both sides of the equation.

$4 x + 32 - 9 x = - 18$

Subtract $9 x$ from $4 x$ .

$- 5 x + 32 = - 18$

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

Subtract $32$ from both sides of the equation.

$- 5 x = - 18 - 32$

Subtract $32$ from $- 18$ .

$- 5 x = - 50$

Divide each term in $- 5 x = - 50$ by $- 5$ and simplify.

Divide each term in $- 5 x = - 50$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 50}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 50}{- 5}$

Divide $x$ by $1$ .

$x = \frac{- 50}{- 5}$

Simplify the right side.

Divide $- 50$ by $- 5$ .

$x = 10$

Solve by Factoring 2 square root of x+8=3 square root of x-2

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