Precalculus Examples

$\frac{4}{1 - \sqrt{5}}$

Step 1

Multiply $\frac{4}{1 - \sqrt{5}}$ by $\frac{1 + \sqrt{5}}{1 + \sqrt{5}}$ .

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

Step 2

Multiply $\frac{4}{1 - \sqrt{5}}$ by $\frac{1 + \sqrt{5}}{1 + \sqrt{5}}$ .

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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

Step 5

Simplify the expression.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Apply the distributive property.

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

Step 7

Multiply $- 1$ by $1$ .

$- 1 - \sqrt{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- 1 - \sqrt{5}$

Decimal Form:

$- 3.23606797 \dots$