# What is the Discriminant?

Mar 9, 2015

$\Delta = {b}^{2} - 4 a c$ for a quadratic $a {x}^{2} + b x + c = 0$

#### Explanation:

The discriminant indicated normally by $\Delta$, is a part of the quadratic formula used to solve second degree equations.
Given a second degree equation in the general form:
$a {x}^{2} + b x + c = 0$
the discriminant is:
$\Delta = {b}^{2} - 4 a c$

The discriminant can be used to characterize the solutions of the equation as:
1) $\Delta > 0$ two separate real solutions;
2) $\Delta = 0$ two coincident real solutions (or one repeated root);
3) $\Delta < 0$ no real solutions.

For example:
${x}^{2} - x - 2 = 0$
Where: $a = 1$, $b = - 1$ and $c = - 2$
So:
$\Delta = {b}^{2} - 4 a c = 1 + 4 \cdot 2 = 9 > 0$, giving $2$ real distinct solutions.

The discriminant can also come in handy when attempting to factorize quadratics. If $\Delta$ is a square number, then the quadratic will factorize, (since the square root in the quadratic formula will be rational). If it is not a square number, then the quadratic will not factorize. This can save you spending ages trying to factorize when it won't work. Instead, solve by completing the square or using the formula.

I hope that helps!

Dec 5, 2017

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.

The discriminant $\Delta$ of a quadratic equation with real coefficients in standard form:

$a {x}^{2} + b x + c = 0$

is given by the formula:

$\Delta = {b}^{2} - 4 a c$

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:

$a {x}^{3} + b {x}^{2} + c x + d = 0$

the discriminant $\Delta$ is given by the formula:

$\Delta = {b}^{2} {c}^{2} - 4 a {c}^{3} - 4 {b}^{3} d - 27 {a}^{2} {d}^{2} + 18 a b c d$

• 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.