Complex Roots of a Quadratic Equation

Complex Roots of a Quadratic Equation

"How to solve the complex roots of a quadratic equation problems: the meaning of the discriminant, examples, and their solutions.

Example 1

Solve the given equation. 4x^2 - x + 5 = 0

Previously, you've solved this example.

Quadratic formula

Let's solve the same equation
by using the quadratic formula
to find its complex roots.

-79 = √79i

Imaginary numbers (i)

So, instead of just writing 'no real roots',
now you can write the roots in complex numbers.

x = (1 ± √79i) / 8.

Meaning of the Discriminant

If D < 0, then the quadratic equation has 2 imaginary roots.

Recall that the discriminant
determines the nature of the roots
of a quadratic equation.

If D < 0, it makes the radicand (-).
So there are no real roots.

This is what you've learned before.

But now you know how to solve (-) radicand:
-1 = i.

Imaginary numbers (i)

So now you can say that
if D < 0,
then the quadratic equation has two imaginary roots.

Example 2

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

D < 0.

So this quadratic equation has two imaginary roots.