Complex Roots of a Quadratic Equation

Complex Roots of a Quadratic Equation

How to solve a quadratic equation that has complex roots and use the discriminant to see if a quadratic equation has complex roots: examples and their solutions.

Example 1: Solve 4x2 - x + 5 = 0

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

Previously, you've solved this example.

Quadratic formula - Example 2

At that time,
the answer was [no real roots].

But, now you know about complex numbers.

So let's find the complex roots
of the given quadratic equation.

a = 4
b = -1
c = 5

So, by the quadratic formula,
x = [-(-1) ± √(-1)2 - 4⋅4⋅5]/[2⋅4].

Quadratic formula

-(-1) = 1

(-1)2 = 1
-4⋅4⋅5 = -80

2⋅4 = 8

1 - 80 = -79

-79 = √79i

Imaginary number (i)

So x = (1 ± √79i)/8.

Meaning of the Discriminant

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

Recall that
if the discriminant D < 0,
then the quadratic equation has [no real roots].

It's true:
it has [two imaginary roots].

The discriminant

Example 2: Discriminant x2 + 2x + 5 = 0

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

a = 1
b = 2
c = 5

So D = 22 - 4⋅1⋅5.

The discriminant

22 = 4
-4⋅1⋅5 = -20

4 - 20 = -16

So D = -16,
which is less than 0.

D < 0

So the given quadratic equation
has two imaginary roots.