Quadratics

[lea.loveland]

I came across the following question on a practice exam. I don't quite understand the explanation of the answer past solving for the two roots.

The two roots to the equation x2 + ax + b = 0 sum to 1 and differ by 5. What is the value of a + b?

Thanks!
Lea

[tommywallach]

Hey Lea,

So the first thing to realize is that roots is just another word for solutions (in case that was unclear). So what we have is a quadratic whose solutions add up to 1 and differ by 5. (As in they are five apart but add up to 1).

x + y = 1

x - y = 5

We can solve these two equations by adding them together.

2x = 6
x = 3

If we plug that back in, we can solve for y = -2.

Now we need to make sure we write the equation correctly. If our SOLUTIONS (roots) are 3 and -2, then our parenthetical version of the quadratic would be: (x - 3) (x + 2) = 0. (because if you plug 3 and -2 into that equation, it solves correctly to zero).

If we factor that out, we get x ^ 2 - x - 6 = 0.

So a = -1 (there's nothing in front of the x but a minus sign, which is the same as -1), and b = -6.

-1 + -6 = -7

Feel free to ask any kind of follow-up question. Thanks!

-t