# What is the Discriminant?

##### 2 Answers

#### Explanation:

The discriminant indicated normally by

Given a second degree equation in the general form:

the discriminant is:

The discriminant can be used to characterize the solutions of the equation as:

1)

2)

3)

For example:

Where:

So:

The discriminant can also come in handy when attempting to factorize quadratics. If

I hope that helps!

See explanation...

#### Explanation:

The discriminant of a polynomial equation is a value computed from the coefficients which helps us determine the type of roots it has - specifically whether they are real or non-real and distinct or repeated.

**Quadratic equations**

The discriminant

#ax^2+bx+c = 0#

is given by the formula:

#Delta = b^2-4ac#

From the discriminant we can discriminate whether the equation has two real roots, one repeated real root or two non-real roots.

- If
#Delta > 0# then the quadratic equation has two distinct real roots. - If
#Delta = 0# then the quadratic equation has one repeated real root. - If
#Delta < 0# then the quadratic equation has no real roots. It has a complex conjugate pair of non-real roots.

**Cubic equations**

For a cubic equation with real coefficients in standard form:

#ax^3+bx^2+cx+d = 0#

the discriminant

#Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd#

- If
#Delta > 0# then the cubic equation has three real roots. - If
#Delta = 0# then the cubic has a repeated root. It may have one real root of multiplicity#3# . Otherwise it may have two distinct real roots, one of which is of multiplicity#2# . - If
#Delta < 0# then the cubic equation has one real root and a complex conjugate pair of complex roots.

**Higher degree**

Polynomial equations of higher degree also have discriminants, which help determine the nature of the roots, but they are less useful for quartics and above.

See https://socratic.org/s/aLqgSvFm for more details.