Algebra Examples

$\frac{2}{- 4 x + 4 \sqrt{5 x}}$

Step 1

Cancel the common factor of $2$ and $- 4 x + 4 \sqrt{5 x}$ .

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Step 1.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{- 4 x + 4 \sqrt{5 x}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $- 4 x$ .

$\frac{2 (1)}{2 (- 2 x) + 4 \sqrt{5 x}}$

Step 1.2.2

Factor $2$ out of $4 \sqrt{5 x}$ .

$\frac{2 (1)}{2 (- 2 x) + 2 (2 \sqrt{5 x})}$

Step 1.2.3

Factor $2$ out of $2 (- 2 x) + 2 (2 \sqrt{5 x})$ .

$\frac{2 (1)}{2 (- 2 x + 2 \sqrt{5 x})}$

Step 1.2.4

Cancel the common factor.

$\frac{2 \cdot 1}{2 (- 2 x + 2 \sqrt{5 x})}$

Step 1.2.5

Rewrite the expression.

$\frac{1}{- 2 x + 2 \sqrt{5 x}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $2$ out of $- 2 x + 2 \sqrt{5 x}$ .

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Step 2.1.1

Factor $2$ out of $- 2 x$ .

$\frac{1}{2 (- x) + 2 \sqrt{5 x}}$

Step 2.1.2

Factor $2$ out of $2 \sqrt{5 x}$ .

$\frac{1}{2 (- x) + 2 (\sqrt{5 x})}$

Step 2.1.3

Factor $2$ out of $2 (- x) + 2 (\sqrt{5 x})$ .

$\frac{1}{2 (- x + \sqrt{5 x})}$

Step 2.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x}$ as ${(5 x)}^{\frac{1}{2}}$ .

$\frac{1}{2 (- x + {(5 x)}^{\frac{1}{2}})}$

Step 2.3

Apply the product rule to $5 x$ .

$\frac{1}{2 (- x + 5^{\frac{1}{2}} x^{\frac{1}{2}})}$

Step 2.4

Factor $x^{\frac{1}{2}}$ out of $- x + 5^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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Step 2.4.1

Factor $x^{\frac{1}{2}}$ out of $- x$ .

$\frac{1}{2 (x^{\frac{1}{2}} (- x^{\frac{1}{2}}) + 5^{\frac{1}{2}} x^{\frac{1}{2}})}$

Step 2.4.2

Factor $x^{\frac{1}{2}}$ out of $5^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$\frac{1}{2 (x^{\frac{1}{2}} (- x^{\frac{1}{2}}) + x^{\frac{1}{2}} \cdot 5^{\frac{1}{2}})}$

Step 2.4.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (- x^{\frac{1}{2}}) + x^{\frac{1}{2}} \cdot 5^{\frac{1}{2}}$ .

$\frac{1}{2 (x^{\frac{1}{2}} (- x^{\frac{1}{2}} + 5^{\frac{1}{2}}))}$

$\frac{1}{2 x^{\frac{1}{2}} (- x^{\frac{1}{2}} + 5^{\frac{1}{2}})}$

Step 3

Simplify with factoring out.

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Step 3.1

Factor $- 1$ out of $- x^{\frac{1}{2}}$ .

$\frac{1}{2 x^{\frac{1}{2}} (- (x^{\frac{1}{2}}) + 5^{\frac{1}{2}})}$

Step 3.2

Factor $- 1$ out of $5^{\frac{1}{2}}$ .

$\frac{1}{2 x^{\frac{1}{2}} (- (x^{\frac{1}{2}}) - 1 (- 5^{\frac{1}{2}}))}$

Step 3.3

Factor $- 1$ out of $- (x^{\frac{1}{2}}) - 1 (- 5^{\frac{1}{2}})$ .

$\frac{1}{2 x^{\frac{1}{2}} (- (x^{\frac{1}{2}} - 5^{\frac{1}{2}}))}$

Step 3.4

Rewrite negatives.

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Step 3.4.1

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

$\frac{1}{2 x^{\frac{1}{2}} (- 1 (x^{\frac{1}{2}} - 5^{\frac{1}{2}}))}$

Step 3.4.2

Move the negative in front of the fraction.

$- \frac{1}{(2 x^{\frac{1}{2}}) (x^{\frac{1}{2}} - 5^{\frac{1}{2}})}$