Math Doubts

Nature of Roots of a Quadratic equation for Negative discriminant

The roots of a quadratic equation are imaginary and distinct if the discriminant of a quadratic equation is negative.


When a quadratic equation is expressed as $ax^2+bx+c = 0$ in algebraic form, the discriminant ($\Delta$ or $D$) of the quadratic equation is written as $b^2-4ac$.

The roots or zeros of the quadratic equation in terms of discriminant are written in the following two forms.

$(1).\,\,\,$ $\dfrac{-b+\sqrt{\Delta}}{2a}$

$(2).\,\,\,$ $\dfrac{-b-\sqrt{\Delta}}{2a}$

If the discriminant of the quadratic equation is negative, then the square root of the discriminant will be undefined. However, the square of a negative quantity can be expressed by an imaginary quantity.

For example $\sqrt{\Delta} \,=\, id$

Now, the zeros or roots of the quadratic equation can be written in the following form.

$(1).\,\,\,$ $\dfrac{-b+id}{2a}$

$(2).\,\,\,$ $\dfrac{-b-id}{2a}$

The two roots clearly reveal that the zeros or roots of the quadratic equation are distinct and imaginary.


$5x^2+7x+6 = 0$

Evaluate the discriminant of this quadratic equation.

$\Delta \,=\, 7^2-4 \times 5 \times 6$

$\implies$ $\Delta \,=\, 49-120$

$\implies$ $\Delta \,=\, -71$

Similarly, find the square root of the discriminant.

$\implies$ $\sqrt{\Delta} \,=\, \sqrt{-71}$

$\implies$ $\sqrt{\Delta} \,=\, i\sqrt{71}$

The zeros or roots of the given quadratic equation are given here.

$\,\,\, \therefore \,\,\,\,\,\,$ $x \,=\, \dfrac{-7+i\sqrt{71}}{10}$ and $x \,=\, \dfrac{-7-i\sqrt{71}}{10}$

Therefore, it is proved that the roots are distinct and complex roots if the discriminant of quadratic equation is less than zero.

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved