We know that a quadratic equation is a second degree polynomial equation of the form ax² + bx + c = 0, where a, b, c are real numbers, x is the unknown variable and a ≠ 0. For the equation ax² + bx + c = 0, the discriminant is given by D = b² – 4ac. It is also denoted by ∆. A quadratic equation has 2 roots. It will be real or imaginary. In this article we discuss the nature of roots depending upon coefficients and discriminant.

If α and β are the values of x which satisfy the quadratic equation, α and β are called the roots of the quadratic equation. Roots are given by the equation (-b±√(b²-4ac))/2a. The nature of the roots depends on the discriminant.

Nature of Roots depending upon Discriminant

According to the value of discriminant, we shall discuss the following cases about the nature of roots.

Case 1: D = 0

If the discriminant is equal to zero (b² – 4ac = 0), a, b, c are real numbers, a≠0, then the roots of the quadratic equation ax² + bx + c = 0, are real and equal. In this case, the roots are x = -b/2a. The graph of the equation touches the X axis at a single point.

Case 2: D > 0

If the discriminant is greater than zero (b² – 4ac > 0), a, b, c are real numbers, a≠0, then the roots of the quadratic equation ax² + bx + c = 0, are real and unequal. The graph of the equation touches the X-axis at two different points.

Case 3: D < 0

If the discriminant is less than zero (b² – 4ac < 0), a, b, c are real numbers, a≠0, then the roots of the quadratic equation ax² + bx + c = 0, are imaginary and unequal. The roots exist in conjugate pairs. The graph of the equation does not touch the X-axis.

Case 4: D > 0 and perfect square

If D > 0 and a perfect square, then the roots of the quadratic equation are real, unequal and rational.

Case 5: D > 0 and not a perfect square

If D > 0 and not a perfect square, then the roots of the quadratic equation are real, unequal and irrational.

We can summarize all the above cases in the table below.

Nature of Roots depending upon coefficients

Depending upon the nature of the coefficients of the quadratic equation, we can summarize the following.

• If c = 0, then one of the roots of the quadratic equation is zero and the other is -b/a.
• If b = c = 0, then both the roots are zero.
• If a = c, then the roots are reciprocal to each other.

Bridge Course – Nature of Roots of Quadratic Equations

Also Read

Quadratic inequalities

Solved Examples

Example 1:

The roots of the quadratic equation 3x²-10x+3 = 0 are

a) real and equal

b) imaginary

c) real, unequal and rational

d) none of these

Solution:

Given equation 3x²-10x+3 = 0

Here discriminant, D = b²-4ac

=> (-10)² – 4×3×3

= 100 – 36

= 64

D is positive and a perfect square.

So the roots of the quadratic equation are real, unequal and rational.

Hence option c is the answer.

Example 2:

Find the value of p if the equation 3x²-18x+p = 0 has real and equal roots.

a) 27

b) 18

c) 9

d) none of these

Solution:

Given 3x²-18x+p = 0 has real and equal roots.

=> b²-4ac = 0

=>(-18)²-4×3×p = 0

=> 324 – 12p = 0

=> p = 324/12

= 27

Hence option a is the answer.

Example 3:

The quadratic equation with real coefficients when one of its root is (3+2i) is

Solution:

Given one root is 3+2i.

Complex roots occur in conjugate pairs.

So other root = 3-2i

Sum of roots = 6

Product of roots = (3+2i)(3-2i) = 13

Required equation is x²-(Sum)x+Product = 0

=> x²-6x+13 = 0

Example 4:

Show that the equation 3x²+4x+6 = 0 has no real roots.

Solution:

Given equation 3x²+4x+6 = 0

Here a = 3, b = 4, c = 6

Discriminant D = b²-4ac

=> 4²-4×3×6

= 16-72

= -56

Since D<0, the roots are imaginary.

Hence the equation has no real roots.

Video Lesson – Nature of Roots