# The Discriminant

How to use the discriminant of a quadratic equation to find the nature of the roots: formula, examples, and their solutions.

## Formula

The qudratic formula's *b*^{2} - 4*ac*

is called the determinant.

It's denoted as *D*.

It's called the determinant because

the *D* determines the nature of the *x* values.

If *D* > 0 and if *D* is a perfect square,

then the radical sign of the formula

will be removed.

( √ ■^{2} = ■ )

So there are 2 rational roots.

If *D* > 0 and if *D* is not a perfect square,

then the radical sign of the formula will remain.

So there are 2 irrational roots.

If *D* = 0,

then the radical part is removed.

So ± sign is gone.

So there's only 1 real root.

If *D* < 0,

it makes the radicand (inside the radical sign) (-).

It doesn't make sense.

So there are no real roots.

## Example 1

*D* > 0 and *D* = 3^{2} (a perfect square).

So this equation has 2 rational roots.

## Example 2

*D* > 0 and *D* is not a perfect square.

So this equation has 2 irrational roots.

## Example 3

*D* = 0.

So this equation has 1 real root.

## Example 4

*D* < 0.

So this equation has no real roots.

If you know about complex numbers,

there's another way to write the answer.

Complex roots of a quadratic equation