Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{1}{2 \sqrt{5} + 2}$ by $\frac{2 \sqrt{5} - 2}{2 \sqrt{5} - 2}$ .

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

Step 1.2

Multiply $\frac{1}{2 \sqrt{5} + 2}$ by $\frac{2 \sqrt{5} - 2}{2 \sqrt{5} - 2}$ .

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

Step 1.3

Expand the denominator using the FOIL method.

$\frac{2 \sqrt{5} - 2}{4 {\sqrt{5}}^{2} - 4 \sqrt{5} + 4 \sqrt{5} - 4} - \frac{1}{2 \sqrt{5} - 2}$

Step 1.4

Simplify.

$\frac{2 \sqrt{5} - 2}{16} - \frac{1}{2 \sqrt{5} - 2}$

Step 1.5

Cancel the common factor of $2 \sqrt{5} - 2$ and $16$ .

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Step 1.5.1

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

$\frac{2 (\sqrt{5}) - 2}{16} - \frac{1}{2 \sqrt{5} - 2}$

Step 1.5.2

Factor $2$ out of $- 2$ .

$\frac{2 (\sqrt{5}) + 2 (- 1)}{16} - \frac{1}{2 \sqrt{5} - 2}$

Step 1.5.3

Factor $2$ out of $2 (\sqrt{5}) + 2 (- 1)$ .

$\frac{2 (\sqrt{5} - 1)}{16} - \frac{1}{2 \sqrt{5} - 2}$

Step 1.5.4

Cancel the common factors.

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Step 1.5.4.1

Factor $2$ out of $16$ .

$\frac{2 (\sqrt{5} - 1)}{2 (8)} - \frac{1}{2 \sqrt{5} - 2}$

Step 1.5.4.2

Cancel the common factor.

$\frac{2 (\sqrt{5} - 1)}{2 \cdot 8} - \frac{1}{2 \sqrt{5} - 2}$

Step 1.5.4.3

Rewrite the expression.

$\frac{\sqrt{5} - 1}{8} - \frac{1}{2 \sqrt{5} - 2}$

Step 1.6

Multiply $\frac{1}{2 \sqrt{5} - 2}$ by $\frac{2 \sqrt{5} + 2}{2 \sqrt{5} + 2}$ .

$\frac{\sqrt{5} - 1}{8} - (\frac{1}{2 \sqrt{5} - 2} \cdot \frac{2 \sqrt{5} + 2}{2 \sqrt{5} + 2})$

Step 1.7

Multiply $\frac{1}{2 \sqrt{5} - 2}$ by $\frac{2 \sqrt{5} + 2}{2 \sqrt{5} + 2}$ .

$\frac{\sqrt{5} - 1}{8} - \frac{2 \sqrt{5} + 2}{(2 \sqrt{5} - 2) (2 \sqrt{5} + 2)}$

Step 1.8

Expand the denominator using the FOIL method.

$\frac{\sqrt{5} - 1}{8} - \frac{2 \sqrt{5} + 2}{4 {\sqrt{5}}^{2} + 4 \sqrt{5} - 4 \sqrt{5} - 4}$

Step 1.9

Simplify.

$\frac{\sqrt{5} - 1}{8} - \frac{2 \sqrt{5} + 2}{16}$

Step 1.10

Cancel the common factor of $2 \sqrt{5} + 2$ and $16$ .

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Step 1.10.1

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

$\frac{\sqrt{5} - 1}{8} - \frac{2 (\sqrt{5}) + 2}{16}$

Step 1.10.2

Factor $2$ out of $2$ .

$\frac{\sqrt{5} - 1}{8} - \frac{2 (\sqrt{5}) + 2 (1)}{16}$

Step 1.10.3

Factor $2$ out of $2 (\sqrt{5}) + 2 (1)$ .

$\frac{\sqrt{5} - 1}{8} - \frac{2 (\sqrt{5} + 1)}{16}$

Step 1.10.4

Cancel the common factors.

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Step 1.10.4.1

Factor $2$ out of $16$ .

$\frac{\sqrt{5} - 1}{8} - \frac{2 (\sqrt{5} + 1)}{2 (8)}$

Step 1.10.4.2

Cancel the common factor.

$\frac{\sqrt{5} - 1}{8} - \frac{2 (\sqrt{5} + 1)}{2 \cdot 8}$

Step 1.10.4.3

Rewrite the expression.

$\frac{\sqrt{5} - 1}{8} - \frac{\sqrt{5} + 1}{8}$

Step 2

Combine the numerators over the common denominator.

$\frac{\sqrt{5} - 1 - (\sqrt{5} + 1)}{8}$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{5} - 1 - \sqrt{5} - 1 \cdot 1}{8}$

Step 3.2

Multiply $- 1$ by $1$ .

$\frac{\sqrt{5} - 1 - \sqrt{5} - 1}{8}$

Step 4

Simplify terms.

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Step 4.1

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

$\frac{0 - 1 - 1}{8}$

Step 4.2

Simplify by subtracting numbers.

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Step 4.2.1

Subtract $1$ from $0$ .

$\frac{- 1 - 1}{8}$

Step 4.2.2

Subtract $1$ from $- 1$ .

$\frac{- 2}{8}$

Step 4.3

Cancel the common factor of $- 2$ and $8$ .

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Step 4.3.1

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{8}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $8$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

$\frac{- 1}{4}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$