Algebra Examples

$\frac{3 + \sqrt{5}}{1 - \sqrt{5}}$

Step 1

Multiply $\frac{3 + \sqrt{5}}{1 - \sqrt{5}}$ by $\frac{1 + \sqrt{5}}{1 + \sqrt{5}}$ .

$\frac{3 + \sqrt{5}}{1 - \sqrt{5}} \cdot \frac{1 + \sqrt{5}}{1 + \sqrt{5}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{3 + \sqrt{5}}{1 - \sqrt{5}}$ by $\frac{1 + \sqrt{5}}{1 + \sqrt{5}}$ .

$\frac{(3 + \sqrt{5}) (1 + \sqrt{5})}{(1 - \sqrt{5}) (1 + \sqrt{5})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(3 + \sqrt{5}) (1 + \sqrt{5})}{1 + \sqrt{5} - \sqrt{5} - {\sqrt{5}}^{2}}$

Step 2.3

Simplify.

$\frac{(3 + \sqrt{5}) (1 + \sqrt{5})}{- 4}$

Step 3

Expand $(3 + \sqrt{5}) (1 + \sqrt{5})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{3 (1 + \sqrt{5}) + \sqrt{5} (1 + \sqrt{5})}{- 4}$

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

$\frac{3 \cdot 1 + 3 \sqrt{5} + \sqrt{5} \cdot 1 + \sqrt{5} \sqrt{5}}{- 4}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $3$ by $1$ .

$\frac{3 + 3 \sqrt{5} + \sqrt{5} \cdot 1 + \sqrt{5} \sqrt{5}}{- 4}$

Step 4.1.2

Multiply $\sqrt{5}$ by $1$ .

$\frac{3 + 3 \sqrt{5} + \sqrt{5} + \sqrt{5} \sqrt{5}}{- 4}$

Step 4.1.3

Combine using the product rule for radicals.

$\frac{3 + 3 \sqrt{5} + \sqrt{5} + \sqrt{5 \cdot 5}}{- 4}$

Step 4.1.4

Multiply $5$ by $5$ .

$\frac{3 + 3 \sqrt{5} + \sqrt{5} + \sqrt{25}}{- 4}$

Step 4.1.5

Rewrite $25$ as $5^{2}$ .

$\frac{3 + 3 \sqrt{5} + \sqrt{5} + \sqrt{5^{2}}}{- 4}$

Step 4.1.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3 + 3 \sqrt{5} + \sqrt{5} + 5}{- 4}$

Step 4.2

Add $3$ and $5$ .

$\frac{8 + 3 \sqrt{5} + \sqrt{5}}{- 4}$

Step 4.3

Add $3 \sqrt{5}$ and $\sqrt{5}$ .

$\frac{8 + 4 \sqrt{5}}{- 4}$

Step 5

Cancel the common factor of $8 + 4 \sqrt{5}$ and $- 4$ .

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Step 5.1

Factor $4$ out of $8$ .

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

Step 5.2

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

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

Step 5.3

Factor $4$ out of $4 (2) + 4 (\sqrt{5})$ .

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

Step 5.4

Move the negative one from the denominator of $\frac{2 + \sqrt{5}}{- 1}$ .

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

Step 6

Rewrite $- 1 \cdot (2 + \sqrt{5})$ as $- (2 + \sqrt{5})$ .

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

Step 7

Apply the distributive property.

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

Step 8

Multiply $- 1$ by $2$ .

$- 2 - \sqrt{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- 2 - \sqrt{5}$

Decimal Form:

$- 4.23606797 \dots$