Refresher - Discriminant

Discriminant

In the above formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case Greek delta, or simply the discriminant: ${\displaystyle \Delta =b^2-4ac}$

A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:

• If the discriminant is positive, then there are two distinct roots, both of which are real numbers: ${\displaystyle \frac{-b+\sqrt{\Delta }}{2a}}$ and ${\displaystyle \frac{-b-\sqrt{\Delta }}{2a}}$

For quadratic equations with integer coefficients, if the discriminant is a perfect square, then the roots are rational numbers—in other cases they may be quadratic irrationals.

• If the discriminant is zero, then there is exactly one distinct real root, sometimes called a double root: ${\displaystyle x=-\frac{b}{2a}}$.
• If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots, which are complex conjugates of each other: ${\displaystyle x=-\frac{b}{2a}+i\frac{\sqrt{\Delta }}{2a},x=-\frac{b}{2a}-i\frac{\sqrt{\Delta }}{2a}}$,where i is the imaginary unit.

Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.

Completing the square

Completing the square is a technique for converting a quadratic polynomial of the form ${\displaystyle a{x}^2+bx+c}$ to the form ${\displaystyle a{(x-h)}^2+k}$.

The result of completing the square may be written as a formula. For the general case: ${\displaystyle a{x}^2+bx+c=a{(x-h)}^2+k}$, where ${\displaystyle h=-\frac{b}{2a}}$ and ${\displaystyle k=c-\frac{b^2}{4a}}$.

Specifically, when ${\displaystyle a=1}$:

${\displaystyle x^2+bx+c={(x-h)}^2+k}$, where ${\displaystyle h=-\frac{b}{2}\text{and}k=c-\frac{b^2}{4}}$