# SOLVING QUADRATIC EQUATIONS WITH COMPLEX ROOTS

A quadratic equation is a second-degree polynomial. A quadratic equation usually has two solutions, which are known as the roots of the equations.

While some quadratic equation has real roots, others have complex root.

In this post, we will carefully examine quadratic equations with complex roots.

Forthwith, let's get started.

A quadratic equation has complex roots if its roots are complex numbers.

For example, the quadratic equation $x^2+2x+5=0$ has a complex number because it roots $-1+2i$ and $-1-2i$ has a complex number.

## How can we solve quadratic equations with complex roots?

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

## How do you know if a quadratic equation has complex roots?

A quadratic equation has a ca complex root if its discriminant is lesser than zero.

The discriminant of a quadratic equation is.
$b^2-4ac$

Therefore, a quadratic equation has a complex root if $b^2-4ac<0$.

Now, let's solve some quadratic equations with complex root

#### Example 1

Solve $2x^2+3x+4$

Solution:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Here, a=2, b=3, c=4

$x=\frac{-3\pm\sqrt{3^2-4(2)(4)}}{2(2)}$

$x=\frac{-3\pm\sqrt{9-32}}{4}$

$x=\frac{-3\pm\sqrt{-23}}{4}$
$x=\frac{-3\pm\sqrt{-1×23}}{4}$

Recalled that $-1=i^2$ according to properties of complex numbers.

$x=\frac{-3\pm\sqrt{i^2×23}}{4}$

$x=\frac{-3\pm(\sqrt{i^2}×\sqrt{23})}{4}$

$x=\frac{-3\pm×(\sqrt{23}i)}{4}$

Splitting the complex number

$x=\frac{-3}{4}\pm\frac{\sqrt{23}i}{4}$

Converting to decimal number

$x=-0.75\pm 1.20i$

Splitting the ± sign

$x=-0.75+1.20i$ or $-0.75-1.20i$

#### Example 2

Solve the values of x in the equation: $-7x-5x-6=0$

Solution:

Here, a=-7, b=-5, c=-6

$x=\frac{-(-5)\pm\sqrt{(-5)^2-4(-7)(-6)}}{2(-7)}$

$x=\frac{-(-5)\pm\sqrt{25-168}}{-14}$

$x=\frac{+5\pm\sqrt{-143}}{-14}$

According to the property of the quadratic equation, $i^2=-1$

$x=\frac{+5\pm\sqrt{143}i}{-14}$

Converting to decimal number

$x=0.36\pm(-0.85)$

Splitting the ± sign

$x=0.36\pm(-0.85)$

$x=0.36+(-0.85)$ or $0.36-(-0.85)$

$x=0.36-0.85$ or $0.36+0.85$

#### Example 3

Solve the quadratic equation $2x^2-3x+9$

Solution:

Here, a=2, b=-3, c=9

$x=\frac{-(-3)\pm\sqrt{(-3)^2-4(2)(9)}}{2(2)}$

$x=\frac{3\pm\sqrt{9-72}}{4}$

$x=\frac{3\pm\sqrt{-63}}{4}$

$x=\frac{3}{4}\pm\frac{\sqrt{-63}}{4}$

$x=\frac{3}{4}\pm\frac{\sqrt{63}i}{4}$

$x=\frac{3}{4}+\frac{\sqrt{63}i}{4}$ or $\frac{3}{4}-\frac{\sqrt{63}i}{4}$

Converting to decimal number

$x=0.75+1.98i$ or $0.75-1.98i$

That will be all for now. We know that solving quadratic equations may seem overwhelming at first, that is why we have created an easy-to-use quadratic equation solver.

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