 Examine the roots of a quadratic equation with the help of the standard form ax² + bx + c = 0 where a, b, c are the real numbers and a ≠0. We can determine the roots of a quadratic equation by using the discriminant formula. Let us study the concept of how to examine the roots of a quadratic equation here. The roots of the quadratic equation can be solved by using the quadratic formula i.e., x = [-b ± √(b² – 4ac)]/2a.

## Examine the Roots of a Quadratic Equation

It is very interesting to examine the roots of a quadratic equation. In the below section the students can learn more about the methods to find the roots of a quadratic equation, the general form of quadratic equation, roots of the quadratic equation, Discriminant to find the nature of roots of the quadratic equation.

Methods to find the roots of a quadratic equation:

There are three methods to find the quadratic equation.
i. Factoring
iii. Complete by squaring

The general form of a quadratic equation:

The general form of the quadratic equation is ax² + bx + c = 0
where a is the coefficient of x² and a ≠ 0
b is the coefficient of x
c is constant
Example: x² + 2x + 1 = 0

The roots of a quadratic equation are nothing but finding the unknown variable x. There are two outcomes of x they may be real or complex numbers. The roots of a quadratic equation help to plot the points on the graph.
If the roots of the quadratic equation are α then the expression will be aα² + bα + c = 0
If the roots of the quadratic equation are β then the expression will be aβ² + bβ + c = 0

Nature of Roots of a Quadratic Equation:

The nature of roots of a quadratic equation is derived from the quadratic formula
x = [-b ± √(b² – 4ac)]/2a
where a, b, c are the real numbers and a ≠ 0.

Discriminant:

The square root part of the quadratic formula (b² – 4ac) is known as the discriminant of the quadratic equation. The discriminant is denoted by D.
D = b² – 4ac
There are three cases in finding the nature of the roots of the equation.
If D = 0 – The equation will have real and equal roots
If D < 0 – The equation will have non-real and unequal roots
If D > 0 – The equation will have real and distinct roots.

Also, See:

Example 1.
Find the roots of the quadratic equation 2x² – x + 1 = 0. whether x = 1 is a solution of this equation or not?
Solution:
Given that
2x² – x + 1 = 0
Substitute x = 1 in the given equation
2(1)² – 1 + 1 = 0
2 – 1 + 1 = 0
2 = 0
Therefore x = 1 is not a solution of the given equation.

Example 2.
Find the value of k for which x = 2 is a root of an equation Kx² – x + 3 = 0.
Solution:
Given that,
Kx² – x + 3 = 0
Sub x = 2 in given equation
K(2)² – 2 + 3 = 0
4K – 2 + 3 = 0
4K + 1 = 0
4K = -1
K = -1/4

Example 3.
Without solving the Quadratic Equation x² – 3x + 1 = 0 find whether x = 1 is a root of this equation or not.
Solution:
Given that,
x² – 3x + 1 = 0
Substitute x = 1 in the given equation
2(1)² – 3(1) + 1 = 0
2 – 3 + 1 = 0
0 = 0
Therefore x = 1 is a solution of the given equation.

Example 4.
Without solving the Quadratic Equation 2x² – 3x + 4 = 0 find whether x = 2 is a root of this equation or not.
Solution:
Given that
2x² – 3x + 4 = 0
Substitute x = 2 in the given equation
2(2)² – 3(1) + 4 = 0
2(4) – 3 + 4 = 0
8 – 3 + 4 = 0
12 – 3 = 0
9 = 0
Therefore x = 1 is not a solution of the given equation.

Example 5.
Find the value of a for which x = 1 is a root of an equation ax² – 2x + 2 = 0.
Solution:
Given that,
ax² – 2x + 2 = 0
Sub x = in given equation
a(1)² – 2(1) + 2 = 0
a – 2 + 2 = 0
a = 0

### FAQs on Examining the Roots of a Quadratic Equation

1. What are the Roots of a Quadratic Equation?

The roots of a quadratic equation ax² + bx + c = 0 are the values of x that satisfy the equation. The solutions of the equation are α and β.

2. What is the Quadratic Formula?

The roots of the quadratic equation can be solved by using the quadratic formula i.e., x = [-b ± √(b² – 4ac)]/2a

3. What is the sum and product of roots of a quadratic equation?

The sum of the roots of a quadratic equation is -b/a
The product of the roots of a quadratic equation is c/a