    # 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.

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.

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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.

## Bridge Course – Nature of Roots of Quadratic Equations ### 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 The discriminant of a quadratic equation is given by D = b2 – 4ac.

If discriminant, D = 0, then the roots are real and equal.

If discriminant, D>0, then the roots are real and unequal.

If discriminant, D<0, then the roots are imaginary and unequal. 