Nature of Roots of Quadratic Equations

The nature of roots of quadratic equations is determined by whether the roots are real, complex, they are equal, or unequal. The nature of the roots of a quadratic equation can be determined by the discriminant of the equation. Let’s learn about that in this post.

What are Roots of a Quadratic Equation

An equation of degree 2 is called a quadratic equation. The general form of a quadratic equation is as follows:

ax²+bx+c = 0 …(1)

where a ≠ 0. Here a, b, and c are real numbers.

The value of x satisfying the above equation (1) is called the roots of the quadratic equation (1).

Methods of Finding the Nature of Roots of Quadratic Equations

Note that there are two roots of a quadratic equation, and they can be determined by Shreedhara Acharya’s formula which are given below:

x = $\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$.

Write D=b²-4ac.

This quantity D=b²-4ac is called the discriminant of the given quadratic equation.

Then, x = (-b+√D)/2a and (-b-√D)/2a.

Depending upon the discriminant D, the nature of the roots of the quadratic equation (1) can be determined as follows:

1. Real and Unequal Roots: If D is greater than 0, then both the roots will be real and unequal. If a, b, and c are rational numbers and D is a perfect square then the roots will be rational numbers; otherwise, they are irrational.
2. Real and Equal Roots: If D is equal to 0, then both the roots will be real and equal and the roots are x=-b, -b.
3. Complex Roots: If D is less than 0, then both the roots will be complex numbers. The roots will be conjugated to each other, and they are x = (-b+i√-D)/2a and (-b-i√-D)/2a.

The above discussion about the nature of the roots of a quadratic equation can be summarized in the table below:

D>0	Real and unequal roots
D=0	Real and equal roots
D<0	Complex roots

Let us now learn the above method with examples to find the nature of the roots of a quadratic equation.

Solved Examples

Example 1: Find the nature of the roots of x² +2x+5=0.

Solution:

Comparing the equation x² +2x+5=0 with ax² +bx+c=0, we get that

a=1, b=2 and c=5

Then the discriminant D=b²-4ac = 2² -4×1×5 = 4-20 = -18<0.

As D<0, the above method says that the roots of x² +2x+5=0 are complex numbers (conjugate to each other).

Have You Read These:

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