How do you simplify (x^2 – 6x + 5)/(x^2 – 25)?

Jun 10, 2016

$\frac{x - 1}{x + 5}$

Explanation:

To simplify we need to factorise both numerator and denominator.

Numerator

We require to find 2 numbers which multiply to give +5 and at the same time sum to give -6 , the coefficient of the x term.

The numbers which multiply are 1 , 5 and -1 ,-5 and the only pair that sum to give -6 are -1 ,-5.

$\Rightarrow {x}^{2} - 6 x + 5 = \left(x - 5\right) \left(x - 1\right)$

Denominator

Here we have a $\textcolor{b l u e}{\text{difference of squares}}$

In general : $\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

here a = x and b = 5

$\Rightarrow {x}^{2} - 25 = \left(x - 5\right) \left(x + 5\right)$

$\Rightarrow \frac{{x}^{2} - 6 x + 5}{{x}^{2} - 25} = \frac{\left(x - 5\right) \left(x - 1\right)}{\left(x - 5\right) \left(x + 5\right)}$

$= \frac{\cancel{\left(x - 5\right)} \left(x - 1\right)}{\cancel{\left(x - 5\right)} \left(x + 5\right)} = \frac{x - 1}{x + 5}$