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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?

 Mar 17, 2021
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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 (in the quadratic).

the value of n + m should be the value of (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.

for example:

2, 27

9, 6

18, 3

 

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 value, though we should be still be careful):

9, 8

6, 12

-9, -8

-3, -24

 

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 smiley

 Mar 17, 2021

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