How to Find Complex Roots of the Quadratic Equation?

The complex roots are a form of complex numbers and are represented as $α = a + ib$, and $β = c + id$. The quadratic equation having a discriminant value lesser than zero $(D<0)$ has imaginary roots, which are represented as complex numbers.

Related Topics

• Identities of Complex Numbers
• How to Solve the Complex Plane
• How to Add and Subtract Complex Numbers

A step-by-step guide to complex roots of the quadratic equation

Complex roots are the imaginary roots of quadratic equations that are represented as complex numbers. The square root of a negative number is not possible and hence we convert it to a complex number. The quadratic equations having discriminant values lesser than zero $b^2-4ac<0$, converted by the use $i^2=-1$, to obtain the complex roots. Here $-D$ is written as $i^2D$.

Complex roots are expressed as complex numbers $a±ib$. The complex root consists of a real part and an imaginary party. Complex roots are often shown as $Z=a+ib$. Here $a$ is the real part of the complex number denoted by Re $(Z)$ and $b$ is the imaginary part denoted by I’m $(Z)$. And $ib$ is the imaginary number.

Note: $i^2= -1$, and the negative number $-N$ is represented as $i^2N$, and it has now transformed into a positive number.

Complex Roots of the Quadratic Equation – Example 1:

Find the complex roots of the quadratic equation $x^2+3x+4=0$.

Solution:

The roots of the quadratic equation $ax^2+bx+c=0$ is equal to $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Here $a=1, b=3,c=4$. Applying this to the formula we have the roots as follows:

$x_{1,2}=\frac{-3\pm \sqrt{3^2-4\times 1\times 4}}{2\times 1}$

$x_{1,2}=\frac{-3\pm \sqrt{9-16}}{2}$

$x_{1,2}=\frac{-3\pm \sqrt{-7}}{2}$

$x_{1,2}=\frac{-3\pm i\sqrt{7}}{2}$

Thus the two complex roots of the quadratic equation are:

$x=\frac{-3+i\sqrt{7}}{2}$ and $x=\frac{-3-i\sqrt{7}}{2}$

Exercises for Complex Roots of the Quadratic Equation

Find the complex roots of the quadratic equation.

1. $\color{blue}{x^2-6x+13=0}$
2. $\color{blue}{3x^2-10x+15=0}$
3. $\color{blue}{x^2+4x+5=0}$
4. $\color{blue}{x^2-3x+10=0}$

1. $\color{blue}{x=3+2i, x=3-2i}$
2. $\color{blue}{x=\frac{5}{3}+\frac{2\sqrt{5}}{3}i,\:x=\frac{5}{3}-\frac{2\sqrt{5}}{3}i}$
3. $\color{blue}{x=-2+i, x=-2-i}$
4. $\color{blue}{x=\frac{3}{2}+\frac{\sqrt{31}}{2}i,\:x=\frac{3}{2}-\frac{\sqrt{31}}{2}i}$