Precalculus Examples

$\sqrt{6 x - 2} = \sqrt{4 x + 4}$

Step 1

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

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

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 $\sqrt{6 x - 2}$ as ${(6 x - 2)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{4 x + 4}}^{2}$ .

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Step 2.3.1.1

Factor $4$ out of $4 x + 4$ .

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Step 2.3.1.1.1

Factor $4$ out of $4 x$ .

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

Step 2.3.1.1.2

Factor $4$ out of $4$ .

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

Step 2.3.1.1.3

Factor $4$ out of $4 (x) + 4 (1)$ .

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

Step 2.3.1.2

Rewrite $4 (x + 1)$ as $2^{2} (x + 1^{2})$ .

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Step 2.3.1.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.3.1.2.2

Rewrite $1$ as $1^{2}$ .

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

Step 2.3.1.3

Pull terms out from under the radical.

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

Step 2.3.1.4

Simplify the expression.

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Step 2.3.1.4.1

One to any power is one.

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

Step 2.3.1.4.2

Apply the product rule to $2 \sqrt{x + 1}$ .

$6 x - 2 = 2^{2} {\sqrt{x + 1}}^{2}$

Step 2.3.1.4.3

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

$6 x - 2 = 4 {\sqrt{x + 1}}^{2}$

Step 2.3.1.5

Rewrite ${\sqrt{x + 1}}^{2}$ as $x + 1$ .

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Step 2.3.1.5.1

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

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

Step 2.3.1.5.2

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

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

Step 2.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 2.3.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.1.5.4.1

Cancel the common factor.

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

Step 2.3.1.5.4.2

Rewrite the expression.

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

Step 2.3.1.5.5

Simplify.

$6 x - 2 = 4 (x + 1)$

Step 2.3.1.6

Apply the distributive property.

$6 x - 2 = 4 x + 4 \cdot 1$

Step 2.3.1.7

Multiply $4$ by $1$ .

$6 x - 2 = 4 x + 4$

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 $4 x$ from both sides of the equation.

$6 x - 2 - 4 x = 4$

Step 3.1.2

Subtract $4 x$ from $6 x$ .

$2 x - 2 = 4$

Step 3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 3.2.1

Add $2$ to both sides of the equation.

$2 x = 4 + 2$

Step 3.2.2

Add $4$ and $2$ .

$2 x = 6$

Step 3.3

Divide each term in $2 x = 6$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 x = 6$ by $2$ .

$\frac{2 x}{2} = \frac{6}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{6}{2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $6$ by $2$ .

$x = 3$