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

ii. Quadratic Formula

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

**Roots of the Quadratic Equation:**

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:

- Methods of Solving Quadratic Equations
- Roots of a Quadratic Equation
- Worksheet on Nature of the Roots of a Quadratic Equation

### Nature of Roots of Quadratic Equation Questions and Answers

**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