Solve by Factoring 2x = square root of 6+5x

$2 x = \sqrt{6 + 5 x}$

Step 1

Subtract $\sqrt{6 + 5 x}$ from both sides of the equation.

$2 x - \sqrt{6 + 5 x} = 0$

Step 2

Subtract $2 x$ from both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 + 5 x}$ as ${(6 + 5 x)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${(- {(6 + 5 x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(6 + 5 x)}^{\frac{1}{2}}$ .

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

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

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

Multiply ${({(6 + 5 x)}^{\frac{1}{2}})}^{2}$ by $1$ .

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

Multiply the exponents in ${({(6 + 5 x)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Apply the product rule to $- 2 x$ .

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

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

$6 + 5 x = 4 x^{2}$

Step 5

Solve for $x$ .

Subtract $4 x^{2}$ from both sides of the equation.

$6 + 5 x - 4 x^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $6 + 5 x - 4 x^{2}$ .

Reorder the expression.

Move $6$ .

$5 x - 4 x^{2} + 6 = 0$

Reorder $5 x$ and $- 4 x^{2}$ .

$- 4 x^{2} + 5 x + 6 = 0$

Factor $- 1$ out of $- 4 x^{2}$ .

$- (4 x^{2}) + 5 x + 6 = 0$

Factor $- 1$ out of $5 x$ .

$- (4 x^{2}) - (- 5 x) + 6 = 0$

Rewrite $6$ as $- 1 (- 6)$ .

$- (4 x^{2}) - (- 5 x) - 1 \cdot - 6 = 0$

Factor $- 1$ out of $- (4 x^{2}) - (- 5 x)$ .

$- (4 x^{2} - 5 x) - 1 \cdot - 6 = 0$

Factor $- 1$ out of $- (4 x^{2} - 5 x) - 1 (- 6)$ .

$- (4 x^{2} - 5 x - 6) = 0$

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 $a \cdot c = 4 \cdot - 6 = - 24$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

$- (4 x^{2} - 5 x - 6) = 0$

Rewrite $- 5$ as $3$ plus $- 8$

$- (4 x^{2} + (3 - 8) x - 6) = 0$

Apply the distributive property.

$- (4 x^{2} + 3 x - 8 x - 6) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- ((4 x^{2} + 3 x) - 8 x - 6) = 0$

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

$- (x (4 x + 3) - 2 (4 x + 3)) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x + 3$ .

$- ((4 x + 3) (x - 2)) = 0$

Remove unnecessary parentheses.

$- (4 x + 3) (x - 2) = 0$

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

$4 x + 3 = 0$

$x - 2 = 0$

Set $4 x + 3$ equal to $0$ and solve for $x$ .

Set $4 x + 3$ equal to $0$ .

$4 x + 3 = 0$

Solve $4 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Divide each term in $4 x = - 3$ by $4$ and simplify.

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $- (4 x + 3) (x - 2) = 0$ true.

$x = - \frac{3}{4}, 2$

Step 6

Exclude the solutions that do not make $2 x - \sqrt{6 + 5 x} = 0$ true.

$x = 2$

Solve by Factoring 2x = square root of 6+5x

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