Nature of Roots





This page explains nature of roots of quadratic equation.

The roots of the quadratic equation ax² + bx+ c = 0 are [- b ± √(b² - 4 ac)]/2a. The nature of roots depends on the value b² - 4 ac. The value of the expression b² - 4 ac discriminates the nature and so it is called the discriminant of the quadratic equation. It is denoted by symbol ∆.

Discriminant ∆= b² - 4 ac

Nature of roots

(1) ∆ > 0 but not a perfect square

Real,unequal and irrational

(2) ∆ > 0 but a perfect square

Real,unequal and rational

(3) ∆ = 0

Real, equal and rational

(4) ∆ < 0 but a perfect square

Unreal (imaginary)

To understand this topic clearly we have shown the following examples

Example 1:

Determine the nature of the roots of the equation x² - 11 x - 30 = 0

Solution:

To get the values of a , b and c we have to compare the given equation with the general form of quadratic equation a x² + b x + c = 0

a = 1  b = -11 and c = -30

Discriminant ∆ = b² - 4 ac

                    = (-11)² - 4 (1) (-30)

                    = 121 + 120

                    = 241

The value of ∆ is 241 that is ∆ > 0 but not a perfect square. Hence the roots are real,unequal and irrational.


Example 2:

Determine the nature of the roots of the equation 5 x² - 2 x - 7 = 0

Solution:

To get the values of a , b and c we have to compare the given equation with the general form of quadratic equation a x² + b x + c = 0

a = 5  b = -2 and c = -7

Discriminant ∆ = b² - 4 ac

                    = (-2)² - 4 (5) (-7)

                    = 4 + 140

                    = 144

The value of ∆ is 144 that is ∆ > 0 but it is a perfect square. Hence the roots are real,unequal and rational.


Example 3:

Determine the nature of the roots of the equation 4 x² - 28 x + 49 = 0

Solution:

To get the values of a , b and c we have to compare the given equation with the general form of quadratic equation a x² + b x + c = 0

a = 4  b = -28 and c = 49

Discriminant ∆ = b² - 4 ac

                    = (-28)² - 4 (4) (49)

                    = 784 - 784

                    = 0

The value of ∆ is 0 that is ∆ = 0.Hence the roots are real, equal and rational.


Example 4:

Determine the nature of the roots of the equation x² - 2 x + 5 = 0

Solution:

To get the values of a , b and c we have to compare the given equation with the general form of quadratic equation a x² + b x + c = 0

a = 1  b = -2 and c = 5

Discriminant ∆ = b² - 4 ac

                    = (-2)² - 4 (1) (5)

                    = 4 - 20

                    = -16

The value of ∆ is 0 that is ∆ <  0 Hence the roots are imaginary.