Question

Find a formula for the sum S of the solutions of the quadratic equation ax²+bx+c=0 .

Answer 1

Answer:

$S = -\frac{b}{a}$

Step-by-step explanation:

Given a second order polynomial expressed by the following equation:

$ax^{2} + bx + c, a\neq0$

This polynomial has roots $x_{1}, x_{2}$ such that $ax^{2} + bx + c = (x - x_{1})*(x - x_{2})$, given by the following formulas:

$x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a}$

$x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a}$

$\bigtriangleup = b^{2} - 4ac$

In this problem, the sum S of the solutions of a quadratic equation is:

$S = x_{1} + x_{2}$

So

$S = \frac{-b + \sqrt{\bigtriangleup}}{2*a} + \frac{-b - \sqrt{\bigtriangleup}}{2*a}$

They have the same denominators, so we can keep the denominators and sum the numerators.

$S = \frac{-b + \sqrt{\bigtriangleup} - b - \sqrt{\bigtriangleup}}{2a}$

$S = \frac{-2b}{2a}$

$S = -\frac{b}{a}$