Algebra Examples

$\sqrt{9 x^{2} + 4 x - 4} = 3 x$

Step 1

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

${\sqrt{9 x^{2} + 4 x - 4}}^{2} = {(3 x)}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 x^{2} + 4 x - 4}$ as ${(9 x^{2} + 4 x - 4)}^{\frac{1}{2}}$ .

${({(9 x^{2} + 4 x - 4)}^{\frac{1}{2}})}^{2} = {(3 x)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(9 x^{2} + 4 x - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(9 x^{2} + 4 x - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(9 x^{2} + 4 x - 4)}^{\frac{1}{2} \cdot 2} = {(3 x)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(9 x^{2} + 4 x - 4)}^{\frac{1}{2} \cdot 2} = {(3 x)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(9 x^{2} + 4 x - 4)}^{1} = {(3 x)}^{2}$

Step 2.2.1.2

Simplify.

$9 x^{2} + 4 x - 4 = {(3 x)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the product rule to $3 x$ .

$9 x^{2} + 4 x - 4 = 3^{2} x^{2}$

Step 2.3.1.2

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

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

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Combine the opposite terms in $9 x^{2} + 4 x - 4 - 9 x^{2}$ .

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Step 3.1.2.1

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

$4 x - 4 + 0 = 0$

Step 3.1.2.2

Add $4 x - 4$ and $0$ .

$4 x - 4 = 0$

Step 3.2

Add $4$ to both sides of the equation.

$4 x = 4$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $4$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $x$ by $1$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $4$ by $4$ .

$x = 1$