Algebra Examples

$\sqrt{4 x - 20} + 3 \sqrt{\frac{x - 5}{9}} - \frac{1}{3} \sqrt{9 x - 45} = 4$

Step 1

Simplify $\sqrt{4 x - 20} + 3 \sqrt{\frac{x - 5}{9}} - \frac{1}{3} \cdot \sqrt{9 x - 45}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $\frac{x - 5}{9}$ as ${(\frac{1}{3})}^{2} (x - 5)$ .

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Step 1.1.1.1

Factor the perfect power $1^{2}$ out of $x - 5$ .

$\sqrt{4 x - 20} + 3 \sqrt{\frac{1^{2} (x - 5)}{9}} - \frac{1}{3} \cdot \sqrt{9 x - 45} = 4$

Step 1.1.1.2

Factor the perfect power $3^{2}$ out of $9$ .

$\sqrt{4 x - 20} + 3 \sqrt{\frac{1^{2} (x - 5)}{3^{2} \cdot 1}} - \frac{1}{3} \cdot \sqrt{9 x - 45} = 4$

Step 1.1.1.3

Rearrange the fraction $\frac{1^{2} (x - 5)}{3^{2} \cdot 1}$ .

$\sqrt{4 x - 20} + 3 \sqrt{{(\frac{1}{3})}^{2} (x - 5)} - \frac{1}{3} \cdot \sqrt{9 x - 45} = 4$

Step 1.1.2

Pull terms out from under the radical.

$\sqrt{4 x - 20} + 3 (\frac{1}{3} \sqrt{x - 5}) - \frac{1}{3} \cdot \sqrt{9 x - 45} = 4$

Step 1.1.3

Combine $\frac{1}{3}$ and $\sqrt{x - 5}$ .

$\sqrt{4 x - 20} + 3 \frac{\sqrt{x - 5}}{3} - \frac{1}{3} \cdot \sqrt{9 x - 45} = 4$

Step 1.1.4

Cancel the common factor of $3$ .

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Step 1.1.4.1

Cancel the common factor.

$\sqrt{4 x - 20} + 3 \frac{\sqrt{x - 5}}{3} - \frac{1}{3} \cdot \sqrt{9 x - 45} = 4$

Step 1.1.4.2

Rewrite the expression.

$\sqrt{4 x - 20} + \sqrt{x - 5} - \frac{1}{3} \cdot \sqrt{9 x - 45} = 4$

Step 1.1.5

Factor $9$ out of $9 x - 45$ .

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Step 1.1.5.1

Factor $9$ out of $9 x$ .

$\sqrt{4 x - 20} + \sqrt{x - 5} - \frac{1}{3} \cdot \sqrt{9 (x) - 45} = 4$

Step 1.1.5.2

Factor $9$ out of $- 45$ .

$\sqrt{4 x - 20} + \sqrt{x - 5} - \frac{1}{3} \cdot \sqrt{9 x + 9 \cdot - 5} = 4$

Step 1.1.5.3

Factor $9$ out of $9 x + 9 \cdot - 5$ .

$\sqrt{4 x - 20} + \sqrt{x - 5} - \frac{1}{3} \cdot \sqrt{9 (x - 5)} = 4$

Step 1.1.6

Rewrite $9$ as $3^{2}$ .

$\sqrt{4 x - 20} + \sqrt{x - 5} - \frac{1}{3} \cdot \sqrt{3^{2} (x - 5)} = 4$

Step 1.1.7

Pull terms out from under the radical.

$\sqrt{4 x - 20} + \sqrt{x - 5} - \frac{1}{3} \cdot (3 \sqrt{x - 5}) = 4$

Step 1.1.8

Cancel the common factor of $3$ .

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Step 1.1.8.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\sqrt{4 x - 20} + \sqrt{x - 5} + \frac{- 1}{3} \cdot (3 \sqrt{x - 5}) = 4$

Step 1.1.8.2

Factor $3$ out of $3 \sqrt{x - 5}$ .

$\sqrt{4 x - 20} + \sqrt{x - 5} + \frac{- 1}{3} \cdot (3 (\sqrt{x - 5})) = 4$

Step 1.1.8.3

Cancel the common factor.

$\sqrt{4 x - 20} + \sqrt{x - 5} + \frac{- 1}{3} \cdot (3 \sqrt{x - 5}) = 4$

Step 1.1.8.4

Rewrite the expression.

$\sqrt{4 x - 20} + \sqrt{x - 5} - 1 \cdot \sqrt{x - 5} = 4$

Step 1.1.9

Rewrite $- 1 \sqrt{x - 5}$ as $- \sqrt{x - 5}$ .

$\sqrt{4 x - 20} + \sqrt{x - 5} - \sqrt{x - 5} = 4$

Step 1.2

Combine the opposite terms in $\sqrt{4 x - 20} + \sqrt{x - 5} - \sqrt{x - 5}$ .

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Step 1.2.1

Subtract $\sqrt{x - 5}$ from $\sqrt{x - 5}$ .

$\sqrt{4 x - 20} + 0 = 4$

Step 1.2.2

Add $\sqrt{4 x - 20}$ and $0$ .

$\sqrt{4 x - 20} = 4$

Step 2

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

${\sqrt{4 x - 20}}^{2} = 4^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x - 20}$ as ${(4 x - 20)}^{\frac{1}{2}}$ .

${({(4 x - 20)}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(4 x - 20)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(4 x - 20)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(4 x - 20)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(4 x - 20)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(4 x - 20)}^{1} = 4^{2}$

Step 3.2.1.2

Simplify.

$4 x - 20 = 4^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $4$ to the power of $2$ .

$4 x - 20 = 16$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Add $20$ to both sides of the equation.

$4 x = 16 + 20$

Step 4.1.2

Add $16$ and $20$ .

$4 x = 36$

Step 4.2

Divide each term in $4 x = 36$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 x = 36$ by $4$ .

$\frac{4 x}{4} = \frac{36}{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{36}{4}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{36}{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $36$ by $4$ .

$x = 9$