User:DTR/Roots of a Quadratic

Introduction edit

A quadratic equation is any equation of the form $ax^{2}+bx+c=0$ . All quadratics have two solutions, or roots. Remember from C1 that these roots may be equal, or not even real! We can also factorise a quadratic, to express it as two linear factors. This chapter is concerned with the roots of various quadratics, and the relationships between the roots.

$x^{2}+7x+10=0$ can be factorised to give $(x+2)(x+5)=0$ . This means the roots are -2 and -5.

Likewise, $x^{2}+6x+9=0$ ⇒ $(x+3)(x+3)=0.$ This means the roots are -3 and -3: a repeated root, because the quadratic is a perfect square.

So that we can refer to and manipulate the roots, we denote them with the Greek letters $\alpha$ and $\beta$ , pronounced alpha and beta, respectively. It can be shown that the sum of the roots, $\alpha +\beta ={\frac {-b}{a}}$ . Also, the product of the roots, $\alpha \beta ={\frac {c}{a}}$ .

General Form edit

The general form of a quadratic with roots $\alpha$ and $\beta$ is $x^{2}-(\alpha +\beta )x+(\alpha \beta )=0$

Or, in a more comprehensible form, $x^{2}-(sum\ of\ roots)x+(product\ of\ roots)=0$ .