Solve by Factoring square root of 8x^2-x=2x

$\sqrt{8 x^{2} - x} = 2 x$

Step 1

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

$\sqrt{8 x^{2} - x} - 2 x = 0$

Step 2

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

Factor $x$ out of $8 x^{2}$ .

$\sqrt{x (8 x) - x} - 2 x = 0$

Factor $x$ out of $- x$ .

$\sqrt{x (8 x) + x \cdot - 1} - 2 x = 0$

Factor $x$ out of $x (8 x) + x \cdot - 1$ .

$\sqrt{x (8 x - 1)} - 2 x = 0$

Step 3

Add $2 x$ to both sides of the equation.

$\sqrt{x (8 x - 1)} = 2 x$

Step 4

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

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

Step 5

Simplify each side of the equation.

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

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

Simplify the left side.

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

Simplify by multiplying through.

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

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

${(x (8 x - 1))}^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x (8 x - 1))}^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

Rewrite the expression.

${(x (8 x - 1))}^{1} = {(2 x)}^{2}$

Apply the distributive property.

${(x (8 x) + x \cdot - 1)}^{1} = {(2 x)}^{2}$

Reorder.

Rewrite using the commutative property of multiplication.

${(8 x \cdot x + x \cdot - 1)}^{1} = {(2 x)}^{2}$

Move $- 1$ to the left of $x$ .

${(8 x \cdot x - 1 \cdot x)}^{1} = {(2 x)}^{2}$

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

${(8 (x \cdot x) - 1 \cdot x)}^{1} = {(2 x)}^{2}$

Multiply $x$ by $x$ .

${(8 x^{2} - 1 \cdot x)}^{1} = {(2 x)}^{2}$

Rewrite $- 1 x$ as $- x$ .

${(8 x^{2} - x)}^{1} = {(2 x)}^{2}$

Simplify.

$8 x^{2} - x = {(2 x)}^{2}$

Simplify the right side.

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

Apply the product rule to $2 x$ .

$8 x^{2} - x = 2^{2} x^{2}$

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

$8 x^{2} - x = 4 x^{2}$

Step 6

Solve for $x$ .

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

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

$8 x^{2} - x - 4 x^{2} = 0$

Subtract $4 x^{2}$ from $8 x^{2}$ .

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

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

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

$x (4 x) - x = 0$

Factor $x$ out of $- x$ .

$x (4 x) + x \cdot - 1 = 0$

Factor $x$ out of $x (4 x) + x \cdot - 1$ .

$x (4 x - 1) = 0$

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

$x = 0$

$4 x - 1 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $4 x - 1$ equal to $0$ and solve for $x$ .

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Solve $4 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$4 x = 1$

Divide each term in $4 x = 1$ by $4$ and simplify.

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{1}{4}$

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

$x = 0, \frac{1}{4}$

Solve by Factoring square root of 8x^2-x=2x

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