Algebra Examples

$\sqrt{x + 4} = 3 \sqrt{x}$

Step 1

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

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

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 $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

${(x + 4)}^{1} = {(3 \sqrt{x})}^{2}$

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Simplify the expression.

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Step 2.3.1.1.1

Apply the product rule to $3 \sqrt{x}$ .

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

Step 2.3.1.1.2

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

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

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

$x + 4 = 9 x^{\frac{1}{2} \cdot 2}$

Step 2.3.1.2.3

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

$x + 4 = 9 x^{\frac{2}{2}}$

Step 2.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.2.4.1

Cancel the common factor.

$x + 4 = 9 x^{\frac{2}{2}}$

Step 2.3.1.2.4.2

Rewrite the expression.

$x + 4 = 9 x^{1}$

Step 2.3.1.2.5

Simplify.

$x + 4 = 9 x$

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 $9 x$ from both sides of the equation.

$x + 4 - 9 x = 0$

Step 3.1.2

Subtract $9 x$ from $x$ .

$- 8 x + 4 = 0$

Step 3.2

Subtract $4$ from both sides of the equation.

$- 8 x = - 4$

Step 3.3

Divide each term in $- 8 x = - 4$ by $- 8$ and simplify.

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Step 3.3.1

Divide each term in $- 8 x = - 4$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 4}{- 8}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 8$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- 4}{- 8}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 8}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $- 4$ and $- 8$ .

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Step 3.3.3.1.1

Factor $- 4$ out of $- 4$ .

$x = \frac{- 4 (1)}{- 8}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $- 4$ out of $- 8$ .

$x = \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 3.3.3.1.2.2

Cancel the common factor.

$x = \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 3.3.3.1.2.3

Rewrite the expression.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.5$