um and product of the roots of a quadratic equation

We learned on the previous page (The Quadratic Formula), in general there are two roots for any quadratic equation \displaystyle{a}{x}^{2}+{b}{x}+{c}={0}ax2+bx+c=0. Let's denote those roots \displaystyle\alphaα and \displaystyle\betaβ, as follows:

\displaystyle\alpha=\frac{{-{b}+\sqrt{{{b}^{2}-{4}{a}{c}}}}}{{{2}{a}}}α=2a−b+b2−4ac​​ and

\displaystyle\beta=\frac{{-{b}-\sqrt{{{b}^{2}-{4}{a}{c}}}}}{{{2}{a}}}β=2a−b−b2−4ac​​

Sum of the roots α and β

We can add \displaystyle\alphaα and \displaystyle\betaβ as follows:

\displaystyle\alpha+\beta=\frac{{-{b}+\sqrt{{{b}^{2}-{4}{a}{c}}}}}{{{2}{a}}}+\frac{{-{b}-\sqrt{{{b}^{2}-{4}{a}{c}}}}}{{{2}{a}}}α+β=2a−b+b2−4ac​​+2a−b−b2−4ac​​

\displaystyle=\frac{{-{2}{b}+{0}}}{{{2}{a}}}=2a−2b+0​

\displaystyle=-\frac{b}{{a}}=−ab​

Product of the roots α and β

We can multiply \displaystyle\alphaα and \displaystyle\betaβ as follows. First, recall that in general,

\displaystyle{\left({X}+{Y}\right)}{\left({X}-{Y}\right)}={X}^{2}-{Y}^{2}(X+Y)(X−Y)=X2−Y2 and

\displaystyle{\left(\sqrt{{{X}}}\right)}^{2}={X}(X​)2=X

We make use of these to obtain:

\displaystyle\alpha\times\beta=\frac{{-{b}+\sqrt{{{b}^{2}-{4}{a}{c}}}}}{{{2}{a}}}\times\frac{{-{b}-\sqrt{{{b}^{2}-{4}{a}{c}}}}}{{{2}{a}}}α×β=2a−b+b2−4ac​​×2a−b−b2−4ac​​

\displaystyle=\frac{{{\left(-{b}\right)}^{2}-{\left(\sqrt{{{b}^{2}-{4}{a}{c}}}\right)}^{2}}}{{\left({2}{a}\right)}^{2}}=(2a)2(−b)2−(b2−4ac​)2​

\displaystyle=\frac{{{b}^{2}-{\left({b}^{2}-{4}{a}{c}\right)}}}{{{4}{a}^{2}}}=4a2b2−(b2−4ac)​

\displaystyle=\frac{{{4}{a}{c}}}{{{4}{a}^{2}}}=4a24ac​

\displaystyle=\frac{c}{{a}}=ac​

Summary

The sum of the roots \displaystyle\alphaα and \displaystyle\betaβ of a quadratic equation are:

\displaystyle\alpha+\beta=-\frac{b}{{a}}α+β=−ab​

The product of the roots \displaystyle\alphaα and \displaystyle\betaβ is given by:

\displaystyle\alpha\beta=\frac{c}{{a}}αβ=ac​

It's also important to realize that if \displaystyle\alphaα and \displaystyle\betaβ are roots, then:

\displaystyle{\left({x}-\alpha\right)}{\left({x}-\beta\right)}={0}(x−α)(x−β)=0

We can expand the left side of the above equation to give us the following form for the quadratic formula:

\displaystyle{x}^{2}-{\left(\alpha+\beta\right)}{x}+\alpha\beta={0}x2−(α+β)x+αβ=0

Let's use these results to solve a few problems.

Example 1

The quadratic equation \displaystyle{2}{x}^{2}-{7}{x}-{5}={0}2x2−7x−5=0 has roots \displaystyle\alphaα and \displaystyle\betaβ. Find: