Simplify the following:
(x^2 + 3 x - 28)/(x^2 - 7 x + 12)
Factor x^2 - 7 x + 12 by finding two numbers whose product is 12 and whose sum is -7.
The factors of 12 that sum to -7 are -3 and -4. So, x^2 - 7 x + 12 = (x - 3) (x - 4):
(x^2 + 3 x - 28)/((x - 3) (x - 4))
Factor x^2 + 3 x - 28 by finding two numbers whose product is -28 and whose sum is 3.
The factors of -28 that sum to 3 are 7 and -4. So, x^2 + 3 x - 28 = (x + 7) (x - 4):
((x + 7) (x - 4))/((x - 3) (x - 4))
Cancel common terms in the numerator and denominator of ((x + 7) (x - 4))/((x - 3) (x - 4)).
((x + 7) (x - 4))/((x - 3) (x - 4)) = (x - 4)/(x - 4)×(x + 7)/(x - 3) = (x + 7)/(x - 3):
=(x + 7) / (x - 3)
Simplify the following:
(x^2 + 3 x - 28)/(x^2 - 7 x + 12)
Factor x^2 - 7 x + 12 by finding two numbers whose product is 12 and whose sum is -7.
The factors of 12 that sum to -7 are -3 and -4. So, x^2 - 7 x + 12 = (x - 3) (x - 4):
(x^2 + 3 x - 28)/((x - 3) (x - 4))
Factor x^2 + 3 x - 28 by finding two numbers whose product is -28 and whose sum is 3.
The factors of -28 that sum to 3 are 7 and -4. So, x^2 + 3 x - 28 = (x + 7) (x - 4):
((x + 7) (x - 4))/((x - 3) (x - 4))
Cancel common terms in the numerator and denominator of ((x + 7) (x - 4))/((x - 3) (x - 4)).
((x + 7) (x - 4))/((x - 3) (x - 4)) = (x - 4)/(x - 4)×(x + 7)/(x - 3) = (x + 7)/(x - 3):
=(x + 7) / (x - 3)
