## Nature of Roots of a Quadratic Equation

In previous section, we have studied that the roots of the equation are given by

A quadratic equation has

**· Two distinct real roots if ****.
**

If we get two distinct real roots and

**· Two equal roots, if ****.
**

If then

i.e.

So, the roots are both

**· No real roots, if **

If then there is no real number whose square is

**Note: **** determines whether the quadratic equation **** has real roots or not, hence **** is called the discriminant of quadratic equation.
**

**It is denoted by D.**

## Solved Problems based on Nature of Roots of a Quadratic Equation

### Problem:

Find the discriminant of the quadratic equation and hence find the nature of its roots.

### Solution:

The given equation is of the form where a = 2, b = – 4 and c = 3. Therefore, the discriminant.

\

So, the given equation has no real roots.

### Problem:

Find the discriminant of the equation and hence find the nature of its roots. Find them, if they are real.

### Solution:

Here and

Therefore, discriminant

Hence, the given quadratic equation has two equal real roots.

The roots are i.e., i.e.,

### Problem:

Find the values of k for which the following equation has equal roots:

### Solution:

We have,

Here, and c = 2

\

Þ

Þ

The given equation will have equal roots, if

Þ Þ k – 12 = 0 or k – 14 = 0

Þ k = 12 or, k = 14

### Problem:

Find the values of k for which the given equation has real roots:

(i) (ii) (iii)

### Solution:

(i) We have

Here, and c = –2

\

The given equation will have real roots, if

(ii) The given equation is

Here, a = 9, b = 3k and c = 4

\

The given equation will have real roots, if

Þ

Þ

Þ

Þ

(iii) The given equation is

Here, and c = 1

Þ

The given equation will have real roots, if

Þ

Þ

### Problem:

Find the values of k for which the equation has no real roots.

### Solution:

The given equation is

Comparing the given equation with we have a = 1, b = 5k, c = 16

The given equation will have no real roots if D < 0

Þ

Þ

Þ [If ab < 0 and a > 0, then b < 0]

Þ [If then –a < x < a]

### Problem:

If – 4 is a root of equation and the equation has equal roots, find the values of p and q.

### Solution:

Since – 4 is a root of we have

Þ

Þ Þ …(i)

Putting p = 3 in equation we have

Equation will have equal roots if D = 0 i.e.

Þ Þ [Using (i)]

Þ 9 – 4q = 0 Þ

Hence, p = 3 and

### Problem:

If the roots of the equation are equal, prove that

### Solution:

The given equation is

Comparing the given equation with

We have, and

For real and equal roots, \

Þ

Þ

Þ

Þ

Þ

or

or

### Problem:

If – 5 is a root of the quadratic equation and the quadratic equation has equal roots, find the value of k.

### Solution:

Since – 5 is a root of the equation Therefore,

Þ

Þ

Þ

Putting in we get

This equation will have equal roots, if

Discriminant = 0

Þ 49 – 4 ´ 7 ´ k = 0

Þ