Math: quadratic equations: the sum of the roots formula

Tutoring math, one discovers idiosyncrasies of equations. The tutor mentions one about quadratics.

The following is according to my understanding.

Thumbing through an old textbook, I encountered the idea that, with a quadratic, its two roots (answers) add up to -b/a. That is, if the two roots are x₁ and x₂, then

x₁ + x₂ = -b/a

This assumes the quadratic equation to be of the form ax²+bx+c=0.

Let’s take the example of x²-x-12=0. Then, a=1, b=-1, and c=-12. The equation factors to (x-4)(x+3)=0, meaning it has roots (aka answers) x=4 or x=-3. At the same time, -b/a = -(-1)/1 = 1. Indeed, it’s true that the roots sum to -b/a: 4+-3 = -(-1)/1 = 1.

This might be helpful in a case where it’s easy to find one of the roots by guessing. Consider 4x²+13x-35=0. One root is -5: 4(-5)²+13(-5)-35=0. Then, since the roots sum to -b/a, one can set the other to r, and write r+-5=-13/4. Then r=5-13/4 = 7/4. From that, it seems the equation factors as (4x-7)(x+5)=0.

Source:

Travers, K.J., Dalton, L.C., Brunner, V.F., Taylor, A.R. (1977). Using Advanced Algebra. Doubleday Canada Limited.