# Discriminant

The **discriminant** of a quadratic equation of the form is the quantity . When are real, this is a notable quantity, because if the discriminant is positive, the equation has two real roots; if the discriminant is negative, the equation has two nonreal roots; and if the discriminant is 0, the equation has a real double root.

## Contents

## Discriminant of polynomials of degree n

The discriminant can tell us something about the roots of a given polynomial of degree with all the coefficients being real. But for polynomials of degree 4 or higher it can be difficult to use it.

### General formula of discriminant

We know that the discriminant of a polynomial is the product of the squares of the differences of the polynomial roots , so,

#### When

Given a polynomial , its discriminant is , which can also be denoted by .

For we have the graph

which has two distinct real roots.

For we have the graph

which has two non-real roots.

And for the case ,

#### When

The discriminant of a polynomial is given by .

Also, the compressed cubic form has discriminant . We can compress a polynomial of degree 3, which also makes possible to us to use Cardano's formula, by doing the substitution on the polynomial .

- If , then at least two of the roots are equal;
- If , then all three roots are real and distinct;
- If , then one of the roots is real and the other two are complex conjugate.

#### When

The quartic polynomial has discriminant

- If , then at least two of the roots are equal;
- If , then the roots are all real or all non-real;
- If , then there are two real roots and two complex conjugate roots.

#### Some properties

For we can say that

- The polynomial has a multiple root if, and only if, ;
- If , with being a positive integer such that , with being the degree of the polynomial, then there are pairs of complex conjugate roots and real roots;
- If , with being a positive integer such that , then there are pairs of complex conjugate roots and real roots.

## Example Problems

### Introductory

- (AMC 12 2005) There are two values of for which the equation has only one solution for . What is the sum of these values of ?

Solution: Since we want the 's where there is only one solution for , the discriminant has to be . . The sum of these values of is .