Solve the Rational Equation for x square root of 3x+12 = square root of x+8

$\sqrt{3 x + 12} = \sqrt{x + 8}$

Step 1

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

${\sqrt{3 x + 12}}^{2} = {\sqrt{x + 8}}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 12}$ as ${(3 x + 12)}^{\frac{1}{2}}$ .

${({(3 x + 12)}^{\frac{1}{2}})}^{2} = {\sqrt{x + 8}}^{2}$

Simplify the left side.

Simplify ${({(3 x + 12)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 x + 12)}^{\frac{1}{2}})}^{2}$ .

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

${(3 x + 12)}^{\frac{1}{2} \cdot 2} = {\sqrt{x + 8}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x + 12)}^{\frac{1}{2} \cdot 2} = {\sqrt{x + 8}}^{2}$

Rewrite the expression.

${(3 x + 12)}^{1} = {\sqrt{x + 8}}^{2}$

Simplify.

$3 x + 12 = {\sqrt{x + 8}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{x + 8}}^{2}$ as $x + 8$ .

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

$3 x + 12 = {({(x + 8)}^{\frac{1}{2}})}^{2}$

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

$3 x + 12 = {(x + 8)}^{\frac{1}{2} \cdot 2}$

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

$3 x + 12 = {(x + 8)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 x + 12 = {(x + 8)}^{\frac{2}{2}}$

Rewrite the expression.

$3 x + 12 = {(x + 8)}^{1}$

Simplify.

$3 x + 12 = x + 8$

Step 3

Solve for $x$ .

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

Subtract $x$ from both sides of the equation.

$3 x + 12 - x = 8$

Subtract $x$ from $3 x$ .

$2 x + 12 = 8$

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

Subtract $12$ from both sides of the equation.

$2 x = 8 - 12$

Subtract $12$ from $8$ .

$2 x = - 4$

Divide each term in $2 x = - 4$ by $2$ and simplify.

Divide each term in $2 x = - 4$ by $2$ .

$\frac{2 x}{2} = \frac{- 4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 4}{2}$

Divide $x$ by $1$ .

$x = \frac{- 4}{2}$

Simplify the right side.

Divide $- 4$ by $2$ .

$x = - 2$

Solve the Rational Equation for x square root of 3x+12 = square root of x+8

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