What is the nature of roots of the quadratic equation if the value of its discriminant is zero?

Examining the roots of a quadratic equation means to see the type of its roots i.e., whether they are real or imaginary, rational or irrational, equal or unequal.

Table of Contents Show

Nature of Roots depending upon Discriminant
Nature of Roots depending upon coefficients
Bridge Course – Nature of Roots of Quadratic Equations
Solved Examples
Video Lesson – Nature of Roots

The nature of the roots of a quadratic equation depends entirely on the value of its discriminant b\(^{2}\) - 4ac.

In a quadratic equation ax\(^{2}\) + bx + c = 0, a ≠ 0 the coefficients a, b and c are real. We know, the roots (solution) of the equation ax\(^{2}\) + bx + c = 0 are given by x = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).

1. If b\(^{2}\) - 4ac = 0 then the roots will be x = \(\frac{-b ± 0}{2a}\) = \(\frac{-b - 0}{2a}\), \(\frac{-b + 0}{2a}\) = \(\frac{-b}{2a}\), \(\frac{-b}{2a}\).

Clearly, \(\frac{-b}{2a}\) is a real number because b and a are real.

Thus, the roots of the equation ax\(^{2}\) + bx + c = 0 are real and equal if b\(^{2}\) – 4ac = 0.

2. If b\(^{2}\) - 4ac > 0 then \(\sqrt{b^{2} - 4ac}\) will be real and non-zero. As a result, the roots of the equation ax\(^{2}\) + bx + c = 0 will be real and unequal (distinct) if b\(^{2}\) - 4ac > 0.

3. If b\(^{2}\) - 4ac < 0, then \(\sqrt{b^{2} - 4ac}\) will not be real because \((\sqrt{b^{2} - 4ac})^{2}\) = b\(^{2}\) - 4ac < 0 and square of a real number always positive.

Thus, the roots of the equation ax\(^{2}\) + bx + c = 0 are not real if b\(^{2}\) - 4ac < 0.

As the value of b\(^{2}\) - 4ac determines the nature of roots (solution), b\(^{2}\) - 4ac is called the discriminant of the quadratic equation.

Definition of discriminant: For the quadratic equation ax\(^{2}\) + bx + c =0, a ≠ 0; the expression b\(^{2}\) - 4ac is called discriminant and is, in general, denoted by the letter ‘D’.

Thus, discriminant D = b\(^{2}\) - 4ac

Note:

Discriminant of ax\(^{2}\) + bx + c = 0	Nature of roots of ax\(^{2}\) + bx + c = 0	Value of the roots of ax\(^{2}\) + bx + c = 0
b\(^{2}\) - 4ac = 0	Real and equal	- \(\frac{b}{2a}\), -\(\frac{b}{2a}\)
b\(^{2}\) - 4ac > 0	Real and unequal	\(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\)
b\(^{2}\) - 4ac < 0	Not real	No real value

When a quadratic equation has two real and equal roots we say that the equation has only one real solution.

Solved examples to examine the nature of roots of a quadratic equation:

1. Prove that the equation 3x\(^{2}\) + 4x + 6 = 0 has no real roots.

Solution:

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

So, the discriminant = b\(^{2}\) - 4ac

= 4\(^{2}\) - 4 ∙ 3 ∙ 6 = 36 - 72 = -56 < 0.

Therefore, the roots of the given equation are not real.

2. Find the value of ‘p’, if the roots of the following quadratic equation are equal (p - 3)x\(^{2}\) + 6x + 9 = 0.

Solution:

For the equation (p - 3)x\(^{2}\) + 6x + 9 = 0;

a = p - 3, b = 6 and c = 9.

Since, the roots are equal

Therefore, b\(^{2}\) - 4ac = 0

⟹ (6)\(^{2}\) - 4(p - 3) × 9 = 0

⟹ 36 - 36p + 108 = 0

⟹ 144 - 36p = 0

⟹ -36p = - 144

⟹ p = \(\frac{-144}{-36}\)

⟹ p = 4

Therefore, the value of p = 4.

3. Without solving the equation 6x\(^{2}\) - 7x + 2 = 0, discuss the nature of its roots.

Solution:

Comparing 6x\(^{2}\) - 7x + 2 = 0 with ax\(^{2}\) + bx + c = 0 we have a = 6, b = -7, c = 2.

Therefore, discriminant = b\(^{2}\) – 4ac = (-7)\(^{2}\) - 4 ∙ 6 ∙ 2 = 49 - 48 = 1 > 0.

Therefore, the roots (solution) are real and unequal.

Note: Let a, b and c be rational numbers in the equation ax\(^{2}\) + bx + c = 0 and its discriminant b\(^{2}\) - 4ac > 0.

If b\(^{2}\) - 4ac is a perfect square of a rational number then \(\sqrt{b^{2} - 4ac}\) will be a rational number. So, the solutions x = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) will be rational numbers. But if b\(^{2}\) – 4ac is not a perfect square then \(\sqrt{b^{2} - 4ac}\) will be an irrational numberand as a result the solutions x = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) will be irrational numbers. In the above example we found that the discriminant b\(^{2}\) – 4ac = 1 > 0 and 1 is a perfect square (1)\(^{2}\). Also 6, -7 and 2 are rational numbers. So, the roots of 6x\(^{2}\) – 7x + 2 = 0 are rational and unequal numbers.

Quadratic Equation

Introduction to Quadratic Equation

Formation of Quadratic Equation in One Variable

Solving Quadratic Equations

General Properties of Quadratic Equation

Methods of Solving Quadratic Equations

Roots of a Quadratic Equation

Examine the Roots of a Quadratic Equation

Problems on Quadratic Equations

Quadratic Equations by Factoring

Word Problems Using Quadratic Formula

Examples on Quadratic Equations

Word Problems on Quadratic Equations by Factoring

Worksheet on Formation of Quadratic Equation in One Variable

Worksheet on Quadratic Formula

Worksheet on Nature of the Roots of a Quadratic Equation

Worksheet on Word Problems on Quadratic Equations by Factoring

9th Grade Math

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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.

Discriminant	Nature of roots
D = 0	Real and equal roots.
D > 0	Real and unequal roots.
D < 0	Unequal and imaginary
D > 0 and perfect square	Real, unequal and rational
D > 0 and not a perfect square	Real, unequal and irrational

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.