# How do you rationalize the denominator and simplify (2-3sqrt5) / (3+2sqrt5)?

Apr 7, 2016

$\frac{- 36 \pm 13 \sqrt{5}}{11}$

#### Explanation:

Given:$\text{ } \frac{2 - 3 \sqrt{5}}{3 + 2 \sqrt{5}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The value of 1 can be written in many forms:

$1 \text{; "2/2"; "(-5)/(-5)"; } \frac{3 - 2 \sqrt{5}}{3 - 2 \sqrt{5}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Using the principle that $\text{ } {a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

Multiply by 1 but in the form $1 = \frac{3 - 2 \sqrt{5}}{3 - 2 \sqrt{5}}$

$\frac{2 - 3 \sqrt{5}}{3 + 2 \sqrt{5}} \times \frac{3 - 2 \sqrt{5}}{3 - 2 \sqrt{5}} \text{ }$The denominator is now of form (a-b)(a+b)

$\frac{\left(2 - 3 \sqrt{5}\right) \left(3 - 2 \sqrt{5}\right)}{{3}^{2} - \left[{\left(2 \sqrt{5}\right)}^{2}\right]}$

$\frac{6 - 4 \sqrt{5} - 9 \sqrt{5} + 6 {\left(\sqrt{5}\right)}^{2}}{9 - 20}$

$\frac{6 - 13 \sqrt{5} + 30}{9 - 20}$

$\frac{36 - 13 \sqrt{5}}{- 11} \text{ Multiply by 1:} \to \frac{- 1}{- 1} \times \frac{36 - 13 \sqrt{5}}{- 11}$

$\frac{13 \sqrt{5} - 36}{11}$

But $\sqrt{5}$ can be either positive or negative.

In that $\left(- 5\right) \times \left(- 5\right) \text{ "=" "(+5)xx(+5)" "=" } + 5$

$\frac{- 36 \pm 13 \sqrt{5}}{11}$