Question

Simplify completely the quantity x squared plus x minus 12 over quantity x squared minus x minus 20 divided by the quantity 3 x squared minus 24 x plus 45 over quantity 12 x squared minus 48 x minus 60

Answer 1

1. We are asked to simplify the expression : $\frac{ x^{2} +x-12}{ x^{2} -x-20}/ \frac{ 3x^{2} -24x+45}{ 12x^{2} -48x-60}$

2. First thing we can do is to flip the second expression, so make the division a multiplication, and also factorize 3 in the numerator and 12 in the denominator of the second expression as follows:

$\frac{ x^{2} +x-12}{ x^{2} -x-20}* \frac{12( x^{2} -4x-5)}{3(x^{2} -8x+15)}$

3. Now factorize each of the quadratric expressions using the following rule:

when we want to factorize $x^{2} +ax+b$, we look for 2 numbers m and n, whose sum is a, and product is b:
for example: in $x^{2} -x-20$, the 2 numbers we are looking for are clearly -5 and 4, because (-5)+4=-1, (-5)*4=-20, so we write the factorized form (x-5)(x+4) 

Now apply the rule to the whole expression:

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

4. Simplify equal terms in the numerator and denominator:

we get: $\frac{4(x+1)}{(x-5)}$

Answer: $\frac{4(x+1)}{(x-5)}$