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 :
`( x^(2) +x-12)/( x^(2) -x-20)/ ( 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:


`( x^(2) +x-12)/( x^(2) -x-20)* (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:


`((x-3)(x+4))/((x-5)(x+4))* (4(x-5)(x+1))/((x-5)(x-3))`

4. Simplify equal terms in the numerator and denominator:

we get:
`(4(x+1))/((x-5))`

Answer:
`(4(x+1))/((x-5))`