# How to Find Complex Roots of the Quadratic Equation?

Complex roots are the imaginary root of quadratic or polynomial functions. In the following guide, you learn how to find complex roots of quadratic equations.

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.

- \(\color{blue}{x^2-6x+13=0}\)
- \(\color{blue}{3x^2-10x+15=0}\)
- \(\color{blue}{x^2+4x+5=0}\)
- \(\color{blue}{x^2-3x+10=0}\)

- \(\color{blue}{x=3+2i, x=3-2i}\)
- \(\color{blue}{x=\frac{5}{3}+\frac{2\sqrt{5}}{3}i,\:x=\frac{5}{3}-\frac{2\sqrt{5}}{3}i}\)
- \(\color{blue}{x=-2+i, x=-2-i}\)
- \(\color{blue}{x=\frac{3}{2}+\frac{\sqrt{31}}{2}i,\:x=\frac{3}{2}-\frac{\sqrt{31}}{2}i}\)

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