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.

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

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$109.99 Original price was: $109.99.$54.99Current price is: $54.99.