When factoring 5x^2-17x+6=0 ,how do you know what to write as the difference for -17x

Question

When asking my math app, it split -17x into -2x-15x, making my equation 5x^2-2x-15x+6=0. I want to know how they found -2x-15x out of all possible ways to split -17x

Answer 1

You are basically factoring 5x²-17x+6 into the general form (ax+b)(cx+d)

This means the following:

a*c=5 - this has only 2 possible combinations: 5,1 -5,-1

b*d=6 - similarly, there are 4 possibilities: 6,1 -6,-1 2,3 -2,-3

Finally, we have

(a*d)+(b*c)=-17 - since the result is negative at least one of a,b,c,d must also be negative, which eliminates a fair number of possibilities. We can also eliminate a,d being some iteration of 5,6 because there is no combination of 1-,3 or -3,1 that would result in any negative number, let alone -17. By similar paring down of possibilities, the answer eventually is revealed.

 

Answer 2

5x^2-17x+6=0

5x^2-2x-15x+6=0

5x^2-15x-2x+6=0

5x(x-3)-2(x-3)=0

(5x-2)(x-3)=0

First of all, your teacher probably did this problem many times. Secondly, your teacher has more experience factoring polynomials so it's much easier to recognize the correct way to break up -17x.

Here's what I do:

5*6=5*2*3

now group the factors so that the products add (or subtract) to 17

5*6=5*2*3=2*(5*3)=2*15 and 2+15=17