# How do you simplify (25-a^2) / (a^2 +a -30)?

Mar 13, 2018

$\frac{5 + a}{- a - 6}$

#### Explanation:

Given the following identities:

${x}^{2} - {y}^{2} = \left(x + y\right) \left(x - y\right)$

${x}^{2} + \left(a + b\right) x + a b = \left(x + a\right) \left(x + b\right)$

Therefore,

$\frac{25 - {a}^{2}}{{a}^{2} + a - 30}$

$= \frac{\left(5 + a\right) \left(5 - a\right)}{\left(a + 6\right) \left(a - 5\right)}$

$= \frac{\left(5 + a\right) \left(5 - a\right)}{\left(a + 6\right) \left(- 1\right) \left(5 - a\right)}$

$= \frac{5 + a}{- a - 6}$

Note: for the second identity, it is rather more on factorization than a real identity. Hence, more practice could yield faster calculation speed and accuracy.

Mar 13, 2018

$- \frac{a + 5}{a + 6}$

#### Explanation:

$\left(1\right) \text{ }$Factorise top and bottom

$\left(a\right) \text{ }$the top with difference of squares

$\left(b\right) \text{ }$the bottom with usual quadratic form

$\left(2\right) \text{ }$then cancel down

$\frac{25 - {a}^{2}}{{a}^{2} + a - 30}$

=((5-a)(5+a))/((a+6)(a-5)

=(-cancel((a-5))(5+a))/((a+6)cancel((a-5))

$= - \frac{a + 5}{a + 6}$

Mar 13, 2018

Using notable identityties. See below

#### Explanation:

First: we note that $25 - {a}^{2} = \left(5 + a\right) \left(5 - a\right)$

Secondly we look for zeros on denominator in order to factorize it

a=(-b+-sqrt(b^2-4ac))/(2a)=(-1+-sqrt(1^2-4·(-30)·17))/2

$a = \frac{- 1 \pm \sqrt{121}}{2}$. This give us two solutions (roots) of denominator expresion $a = - 6$ and $a = 5$. For this reason we can express denominator as #(a+6)(a-5).

Summarizing all results, we have

$\frac{\left(5 + a\right) \left(5 - a\right)}{\left(a + 6\right) \left(a - 5\right)} = - \frac{\left(5 + a\right) \left(\cancel{a - 5}\right)}{\left(a + 6\right) \left(\cancel{a - 5}\right)} = - \frac{5 + a}{a + 6}$