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.