# 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 4*x*^{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 = √79*i*

Imaginary number (*i*)

So *x* = (1 ± √79*i*)/8.

## Meaning of the Discriminant

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 *x*^{2} + 2*x* + 5 = 0

*a* = 1*b* = 2*c* = 5

So *D* = 2^{2} - 4⋅1⋅5.

The discriminant

2^{2} = 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.