Simplify 5/(-3+ square root of 5)

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

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

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

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

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

Expand the denominator using the FOIL method.

$\frac{5 (- 3 - \sqrt{5})}{9 + 3 \sqrt{5} + \sqrt{5} \cdot - 3 - {\sqrt{5}}^{2}}$

Simplify.

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

Rewrite $- 3$ as $- 1 (3)$ .

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

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

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

Factor $- 1$ out of $- 1 (3) - (\sqrt{5})$ .

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

Move the negative in front of the fraction.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 6.54508497 \dots$