Solve by Factoring 6x = square root of 24+12x

$6 x = \sqrt{24 + 12 x}$

Step 1

Subtract $\sqrt{24 + 12 x}$ from both sides of the equation.

$6 x - \sqrt{24 + 12 x} = 0$

Step 2

Simplify each term.

Factor $12$ out of $24 + 12 x$ .

Factor $12$ out of $24$ .

$6 x - \sqrt{12 \cdot 2 + 12 x} = 0$

Factor $12$ out of $12 \cdot 2 + 12 x$ .

$6 x - \sqrt{12 (2 + x)} = 0$

Rewrite $12 (2 + x)$ as $2^{2} \cdot (3 (2 + x))$ .

Factor $4$ out of $12$ .

$6 x - \sqrt{4 (3) (2 + x)} = 0$

Rewrite $4$ as $2^{2}$ .

$6 x - \sqrt{2^{2} \cdot 3 (2 + x)} = 0$

Add parentheses.

$6 x - \sqrt{2^{2} \cdot (3 (2 + x))} = 0$

Pull terms out from under the radical.

$6 x - (2 \sqrt{3 (2 + x)}) = 0$

Multiply $2$ by $- 1$ .

$6 x - 2 \sqrt{3 (2 + x)} = 0$

Step 3

Factor $2$ out of $6 x - 2 \sqrt{3 (2 + x)}$ .

Factor $2$ out of $6 x$ .

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

Factor $2$ out of $- 2 \sqrt{3 (2 + x)}$ .

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

Factor $2$ out of $2 (3 x) + 2 (- \sqrt{3 (2 + x)})$ .

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

Step 4

Divide each term in $2 (3 x - \sqrt{3 (2 + x)}) = 0$ by $2$ and simplify.

Divide each term in $2 (3 x - \sqrt{3 (2 + x)}) = 0$ by $2$ .

$\frac{2 (3 x - \sqrt{3 (2 + x)})}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (3 x - \sqrt{3 (2 + x)})}{2} = \frac{0}{2}$

Divide $3 x - \sqrt{3 (2 + x)}$ by $1$ .

$3 x - \sqrt{3 (2 + x)} = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

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

Step 5

Subtract $3 x$ from both sides of the equation.

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

Step 6

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

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

Step 7

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the distributive property.

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

Simplify the expression.

Multiply $3$ by $2$ .

${(- {(6 + 3 x)}^{\frac{1}{2}})}^{2} = {(- 3 x)}^{2}$

Apply the product rule to $- {(6 + 3 x)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(6 + 3 x)}^{\frac{1}{2}})}^{2} = {(- 3 x)}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {({(6 + 3 x)}^{\frac{1}{2}})}^{2} = {(- 3 x)}^{2}$

Multiply ${({(6 + 3 x)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(6 + 3 x)}^{\frac{1}{2}})}^{2} = {(- 3 x)}^{2}$

Multiply the exponents in ${({(6 + 3 x)}^{\frac{1}{2}})}^{2}$ .

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

${(6 + 3 x)}^{\frac{1}{2} \cdot 2} = {(- 3 x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(6 + 3 x)}^{\frac{1}{2} \cdot 2} = {(- 3 x)}^{2}$

Rewrite the expression.

${(6 + 3 x)}^{1} = {(- 3 x)}^{2}$

Simplify.

$6 + 3 x = {(- 3 x)}^{2}$

Simplify the right side.

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

Apply the product rule to $- 3 x$ .

$6 + 3 x = {(- 3)}^{2} x^{2}$

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

$6 + 3 x = 9 x^{2}$

Step 8

Solve for $x$ .

Subtract $9 x^{2}$ from both sides of the equation.

$6 + 3 x - 9 x^{2} = 0$

Factor the left side of the equation.

Factor $- 3$ out of $6 + 3 x - 9 x^{2}$ .

Reorder the expression.

Move $6$ .

$3 x - 9 x^{2} + 6 = 0$

Reorder $3 x$ and $- 9 x^{2}$ .

$- 9 x^{2} + 3 x + 6 = 0$

Factor $- 3$ out of $- 9 x^{2}$ .

$- 3 (3 x^{2}) + 3 x + 6 = 0$

Factor $- 3$ out of $3 x$ .

$- 3 (3 x^{2}) - 3 (- x) + 6 = 0$

Factor $- 3$ out of $6$ .

$- 3 (3 x^{2}) - 3 (- x) - 3 \cdot - 2 = 0$

Factor $- 3$ out of $- 3 (3 x^{2}) - 3 (- x)$ .

$- 3 (3 x^{2} - x) - 3 \cdot - 2 = 0$

Factor $- 3$ out of $- 3 (3 x^{2} - x) - 3 (- 2)$ .

$- 3 (3 x^{2} - x - 2) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- x$ .

$- 3 (3 x^{2} - (x) - 2) = 0$

Rewrite $- 1$ as $2$ plus $- 3$

$- 3 (3 x^{2} + (2 - 3) x - 2) = 0$

Apply the distributive property.

$- 3 (3 x^{2} + 2 x - 3 x - 2) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- 3 ((3 x^{2} + 2 x) - 3 x - 2) = 0$

Factor out the greatest common factor (GCF) from each group.

$- 3 (x (3 x + 2) - (3 x + 2)) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 2$ .

$- 3 ((3 x + 2) (x - 1)) = 0$

Remove unnecessary parentheses.

$- 3 (3 x + 2) (x - 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 x + 2 = 0$

$x - 1 = 0$

Set $3 x + 2$ equal to $0$ and solve for $x$ .

Set $3 x + 2$ equal to $0$ .

$3 x + 2 = 0$

Solve $3 x + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$3 x = - 2$

Divide each term in $3 x = - 2$ by $3$ and simplify.

Divide each term in $3 x = - 2$ by $3$ .

$\frac{3 x}{3} = \frac{- 2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 2}{3}$

Divide $x$ by $1$ .

$x = \frac{- 2}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $- 3 (3 x + 2) (x - 1) = 0$ true.

$x = - \frac{2}{3}, 1$

Step 9

Exclude the solutions that do not make $2 (3 x - \sqrt{3 (2 + x)}) = 0$ true.

$x = 1$

Solve by Factoring 6x = square root of 24+12x

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