Algebra Examples

$\sqrt{9 x^{2} - x} = 2 x$

Step 1

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

$\sqrt{9 x^{2} - x} - 2 x = 0$

Step 2

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

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Step 2.1

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

$\sqrt{x (9 x) - x} - 2 x = 0$

Step 2.2

Factor $x$ out of $- x$ .

$\sqrt{x (9 x) + x \cdot - 1} - 2 x = 0$

Step 2.3

Factor $x$ out of $x (9 x) + x \cdot - 1$ .

$\sqrt{x (9 x - 1)} - 2 x = 0$

Step 3

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

$\sqrt{x (9 x - 1)} = 2 x$

Step 4

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

${\sqrt{x (9 x - 1)}}^{2} = {(2 x)}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x (9 x - 1)}$ as ${(x (9 x - 1))}^{\frac{1}{2}}$ .

${({(x (9 x - 1))}^{\frac{1}{2}})}^{2} = {(2 x)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${({(x (9 x - 1))}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Simplify by multiplying through.

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Step 5.2.1.1.1

Multiply the exponents in ${({(x (9 x - 1))}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1.1.1

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

${(x (9 x - 1))}^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

Step 5.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.1.2.1

Cancel the common factor.

${(x (9 x - 1))}^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

Step 5.2.1.1.1.2.2

Rewrite the expression.

${(x (9 x - 1))}^{1} = {(2 x)}^{2}$

Step 5.2.1.1.2

Apply the distributive property.

${(x (9 x) + x \cdot - 1)}^{1} = {(2 x)}^{2}$

Step 5.2.1.1.3

Reorder.

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Step 5.2.1.1.3.1

Rewrite using the commutative property of multiplication.

${(9 x \cdot x + x \cdot - 1)}^{1} = {(2 x)}^{2}$

Step 5.2.1.1.3.2

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

${(9 x \cdot x - 1 \cdot x)}^{1} = {(2 x)}^{2}$

Step 5.2.1.2

Simplify each term.

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Step 5.2.1.2.1

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

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Step 5.2.1.2.1.1

Move $x$ .

${(9 (x \cdot x) - 1 \cdot x)}^{1} = {(2 x)}^{2}$

Step 5.2.1.2.1.2

Multiply $x$ by $x$ .

${(9 x^{2} - 1 \cdot x)}^{1} = {(2 x)}^{2}$

Step 5.2.1.2.2

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

${(9 x^{2} - x)}^{1} = {(2 x)}^{2}$

Step 5.2.1.3

Simplify.

$9 x^{2} - x = {(2 x)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

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

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Step 5.3.1.1

Apply the product rule to $2 x$ .

$9 x^{2} - x = 2^{2} x^{2}$

Step 5.3.1.2

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

$9 x^{2} - x = 4 x^{2}$

Step 6

Solve for $x$ .

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Step 6.1

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

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Step 6.1.1

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

$9 x^{2} - x - 4 x^{2} = 0$

Step 6.1.2

Subtract $4 x^{2}$ from $9 x^{2}$ .

$5 x^{2} - x = 0$

Step 6.2

Factor $x$ out of $5 x^{2} - x$ .

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Step 6.2.1

Factor $x$ out of $5 x^{2}$ .

$x (5 x) - x = 0$

Step 6.2.2

Factor $x$ out of $- x$ .

$x (5 x) + x \cdot - 1 = 0$

Step 6.2.3

Factor $x$ out of $x (5 x) + x \cdot - 1$ .

$x (5 x - 1) = 0$

Step 6.3

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

$x = 0$

$5 x - 1 = 0$

Step 6.4

Set $x$ equal to $0$ .

$x = 0$

Step 6.5

Set $5 x - 1$ equal to $0$ and solve for $x$ .

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Step 6.5.1

Set $5 x - 1$ equal to $0$ .

$5 x - 1 = 0$

Step 6.5.2

Solve $5 x - 1 = 0$ for $x$ .

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Step 6.5.2.1

Add $1$ to both sides of the equation.

$5 x = 1$

Step 6.5.2.2

Divide each term in $5 x = 1$ by $5$ and simplify.

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Step 6.5.2.2.1

Divide each term in $5 x = 1$ by $5$ .

$\frac{5 x}{5} = \frac{1}{5}$

Step 6.5.2.2.2

Simplify the left side.

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Step 6.5.2.2.2.1

Cancel the common factor of $5$ .

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Step 6.5.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{1}{5}$

Step 6.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{5}$

Step 6.6

The final solution is all the values that make $x (5 x - 1) = 0$ true.

$x = 0, \frac{1}{5}$