Solve for x 4x = square root of 6-4x

$4 x = \sqrt{6 - 4 x}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{6 - 4 x} = 4 x$

Step 2

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

${\sqrt{6 - 4 x}}^{2} = {(4 x)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 - 4 x}$ as ${(6 - 4 x)}^{\frac{1}{2}}$ .

${({(6 - 4 x)}^{\frac{1}{2}})}^{2} = {(4 x)}^{2}$

Simplify the left side.

Simplify ${({(6 - 4 x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(6 - 4 x)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(6 - 4 x)}^{1} = {(4 x)}^{2}$

Simplify.

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

Simplify the right side.

Simplify ${(4 x)}^{2}$ .

Apply the product rule to $4 x$ .

$6 - 4 x = 4^{2} x^{2}$

Raise $4$ to the power of $2$ .

$6 - 4 x = 16 x^{2}$

Step 4

Solve for $x$ .

Subtract $16 x^{2}$ from both sides of the equation.

$6 - 4 x - 16 x^{2} = 0$

Factor the left side of the equation.

Factor $- 2$ out of $6 - 4 x - 16 x^{2}$ .

Reorder the expression.

Move $6$ .

$- 4 x - 16 x^{2} + 6 = 0$

Reorder $- 4 x$ and $- 16 x^{2}$ .

$- 16 x^{2} - 4 x + 6 = 0$

Factor $- 2$ out of $- 16 x^{2}$ .

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

Factor $- 2$ out of $- 4 x$ .

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

Factor $- 2$ out of $6$ .

$- 2 (8 x^{2}) - 2 (2 x) - 2 \cdot - 3 = 0$

Factor $- 2$ out of $- 2 (8 x^{2}) - 2 (2 x)$ .

$- 2 (8 x^{2} + 2 x) - 2 \cdot - 3 = 0$

Factor $- 2$ out of $- 2 (8 x^{2} + 2 x) - 2 (- 3)$ .

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

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 $a \cdot c = 8 \cdot - 3 = - 24$ and whose sum is $b = 2$ .

Factor $2$ out of $2 x$ .

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

Rewrite $2$ as $- 4$ plus $6$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$- 2 (4 x (2 x - 1) + 3 (2 x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$- 2 ((2 x - 1) (4 x + 3)) = 0$

Remove unnecessary parentheses.

$- 2 (2 x - 1) (4 x + 3) = 0$

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

$2 x - 1 = 0$

$4 x + 3 = 0$

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

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.

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

The final solution is all the values that make $- 2 (2 x - 1) (4 x + 3) = 0$ true.

$x = \frac{1}{2}, - \frac{3}{4}$

Step 5

Exclude the solutions that do not make $4 x = \sqrt{6 - 4 x}$ true.

$x = \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}$

Decimal Form:

$x = 0.5$

Solve for x 4x = square root of 6-4x

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