Nature of the Roots of the Quadratic Equation - Embibe
• Written By sandeep
• Written By sandeep

# Nature of the Roots of the Quadratic Equation

In mathematics, a quadratic equation is any polynomial equation involving only degree 2, with no linear or higher-order terms. These polynomial equations have the form of $$a{x^2} + bx + c = 0,$$ where $$(a≠0)$$. The roots of the quadratic equation are an algebraic expression that defines the x-intercepts of a quadratic function. The nature of the roots of the quadratic equation is that when these expressions are plotted on a graph, they intersect at two distinct points that defines the x-intercepts of a quadratic function.

The roots of a quadratic equation are the solutions that satisfy the equation. In other words, if we have a quadratic equation with real solutions, then the roots of this equation will be real numbers. However, the way to find the roots of a quadratic equation is by factoring it into two other equations with one unknown and one constant. Then we solve these two equations to find the value of our unknown roots.

## What is a Quadratic Equation?

A polynomial equation of the second degree in x is called a quadratic equation. Such equations have the standard form $$a{x^2} + bx + c = 0$$, where a and b are coefficients, $$x$$ is the variable, and $$c$$ is the constant term. The coefficient of $$x^2$$ is a non-zero term $$(a≠0)$$ and is the first measure for determining whether an equation is quadratic.

Let us take a look at how to find the roots (α, β) of the quadratic equation. First, we need to look at the general formula for solving quadratic equations. The alpha (α) and beta (β) symbols stand for representing the roots of a quadratic equation. The following quadratic equation formula helps in solving and finding the roots of those quadratic equations which are difficult to factorise:

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

The expression under the square root in the above formula is known as Discriminant. The Discriminant, represented by $$D or Δ$$ in the quadratic formula. It helps determine the nature of the roots of the quadratic roots.

### Nature of the Roots of the Quadratic Equation

The value of the Discriminant, $$D=({b^2} – 4ac)$$, determines the nature of the roots of the quadratic equation. If $$a, b, c ∈ R$$, where $$R$$ belongs to real numbers, then the roots of the second-degree equation can either be real or imaginary according to the following criteria:

1. Two distinct real roots, if $${b^2} – 4ac > 0$$
2. Two equal real roots, if $${b^2} – 4ac = 0$$
3. No real roots if $${b^2} – 4ac < 0$$

### How to Effectively Practise?

Are you looking to learn these concepts of Mathematics effectively? The Embibe app is an AI-enabled platform that provides a personalised learning experience. Our platform helps learners with automated assessments, adaptive content, and personalised learning journeys. Above all, it helps learners focus on their weak points by providing targeted practise tests and solutions to their queries.

The app also offers instant feedback on the learners’ performance in the form of grades which are calculated based on the accuracy of responses provided by the learners during tests.

We hope this article on the Nature of the Roots of the Quadratic Equation helps you. If you have any academic queries, do reach out to us, you can email us at support@embibe.com or call 1800313002020 (Toll-Free). We will be happy to help you.

Follow Embibe for more intriguing articles like these. Happy Learning!

Achieve Your Best With 3D Learning, Book Practice, Tests & Doubt Resolutions at Embibe