The Discriminant

The Discriminant

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

Formula

For ax^2 + bx + c = 0, D = b^2 - 4ac. The discriminant determines the nature of the x value.

The qudratic formula's b2 - 4ac
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

Determine the nature of the roots for the given quadratic equation. x^2 + 7x + 10 = 0

D > 0 and D = 32 (a perfect square).

So this equation has 2 rational roots.

Example 2

Determine the nature of the roots for the given quadratic equation. x^2 - 4x - 1 = 0

D > 0 and D is not a perfect square.

So this equation has 2 irrational roots.

Example 3

Determine the nature of the roots for the given quadratic equation. x^2 - 6x + 9 = 0

D = 0.

So this equation has 1 real root.

Example 4

Determine the nature of the roots for the given quadratic equation. x^2 + 2x + 5 = 0

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