The expression x^2 + 15x + 54 can be written as (x + a)(x + b), and the expression x^2 - 17x + 72 written as (x - b)(x - c), where a, b, and c are integers. What is the value of a+b+c?
when factoring quadratics like those (in the form ax^2 + bx + c), here are a few helpful things to keep in mind:
[let's assume that you are factoring the quadratic into something like (x + n)(x + m), where n and m are both constants.
the value of n * m should be the value of c (in the quadratic).
the value of n + m should be the value of b (in the quadratic).
now, let's try and factor your first expression, keeping those in mind.
we can start by listing a few factor pairs of 54.
we can quickly see which ones add up to the b value. now we take these and plug them in to the other expression that was given.
our factored result for the first expression is (x + 6)(x + 9) - a = 6 and b = 9. feel free to expand these to double check.
now, let's factor the next expression. we can list out a few factor pairs of 72. an important thing to remember: negatives also work! don't disregard them.
here is a short list of factor pairs (for your specific problem, we probably didn't even need to do this, because we already know our b value, though we should be still be careful):
the pair that adds up to -17 is -9 and -8. now we can plug those in to the factored expression, to get:
(x - 9)(x - 8).
a = 6
b = 9
c = 8
a + b + c = 23
hope this helped! please let me know if you are still confused