Question

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

Answer 1

`(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.

`rArrx^2-6x+5=(x-5)(x-1)`

Denominator

Here we have a `color(blue)"difference of squares"`

In general : `color(red)(|bar(ul(color(white)(a/a)color(black)(a^2-b^2=(a-b)(a+b))color(white)(a/a)|)))`

here a = x and b = 5

`rArrx^2-25=(x-5)(x+5)`

`rArr(x^2-6x+5)/(x^2-25)=((x-5)(x-1))/((x-5)(x+5))`

`=(cancel((x-5)) (x-1))/(cancel((x-5)) (x+5))=(x-1)/(x+5)`