Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

$\frac{5 (- 4 - \sqrt{5})}{16 + 4 \sqrt{5} + \sqrt{5} \cdot - 4 - {\sqrt{5}}^{2}}$

Step 4

Simplify.

$\frac{5 (- 4 - \sqrt{5})}{11}$

Step 5

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{5 (- 1 (4) - \sqrt{5})}{11}$

Step 6

Factor $- 1$ out of $- \sqrt{5}$ .

$\frac{5 (- 1 (4) - (\sqrt{5}))}{11}$

Step 7

Factor $- 1$ out of $- 1 (4) - (\sqrt{5})$ .

$\frac{5 (- 1 (4 + \sqrt{5}))}{11}$

Step 8

Move the negative in front of the fraction.

$- \frac{5 (4 + \sqrt{5})}{11}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{5 (4 + \sqrt{5})}{11}$

Decimal Form:

$- 2.83457635 \dots$