# Nature of roots of a Quadratic Equation

### Discriminant

= b² -4ac

**Case I.** When a, b, c are real numbers, a 0:

- If = b² -4 a c = 0, then roots are equal (and real).
- If = b² -4 a c > 0, then roots are real and unequal.
- If = b² -4 a c < 0, then roots are complex. It is easy to see that roots are a pair of complex conjugates.

**Case II.** When a, b, c are rational numbers, a 0:

- If = b² -4 a c = 0, then roots are rational and equal.
- If = b² -4 a c > 0, and is a perfect square of a rational number, then roots are rational and unequal.
- If = b² -4 a c > 0 but is not a square of rational number, then roots are irrational and unequal. They form a pair of irrational conjugates p +q, p - q where p, q Q, q> 0.
- If = b² -4 a c < 0, then roots are a pair of complex conjugates.

## Illustrative Examples

### Example

Discuss the nature of the roots of the following equations:

(i) 4 x² -12 x +9 = 0

(ii) 3 x² -10 x +3 = 0

(iii) 9 x² -2 = 0

(iv) x² +x +1 = 0

### Solution

- Here coefficients are rational, and discriminant

= b² -4 a c = (-12)² -4 (4)(9) = 144 -144 = 0.

Hence the roots are rational and equal. - Here coefficients are rational, and discriminant

= b² -4 a c = (-10)² -4 (3)(3) = 100 -36 = 64.

Now = 64 > 0, and 64 is a perfect square of a rational number.

Hence the roots are rational and unequal. - Here coefficients are rational, and discriminant

= b² -4 a c = (0)² -4 (9)(-2) = 72.

Now = 72 > 0 but is not a perfect square of a rational number.

Hence the roots are irrational and unequal. - Here coefficients are rational, and discriminant

= b² -4 a c = (1)² -4 (1)(1) = -3 < 0.

Hence the roots are a pair of complex conjugates.

### Example

Discuss the nature of the roots of the equation

(m +6) x² +(m +6) x +2 = 0

### Solution

Discriminant = (m +6)² -4 (m +6)(2)

= m² +12 m +36 -8 m -48

= m² +4 m -12 = (m +6)(m -2)

- Roots are real and equal if

= (m +6)(m -2) = 0 i.e. if m = 2

*(Ignoring m= -6, as then equation becomes 2=0)* - Roots are real and unequal when

= (m +6)(m -2) > 0 i.e. when m < -6 or when m > 2 - Roots are a pair of complex conjugates when

= (m +6)(m -2) < 0 i.e. when -6 < m < 2

## Exercise

- Find the nature of roots of the following equations without solving them:

(i) x² +9 = 0

(ii) 4 x² -24 x +35 = 0

(iii) x² -22 x +1 = 0

(iv) 2 x² -25 x +3 = 0 - Show that roots of the equation (x -a)(x -b) = a b x² where a, b
R are always real. When are they equal?

[**Hint.**= (a -b)² +(2 a b)²]. - Show that the roots of the equation (x -a)(x -b) +(x -b)(x -c) +(x -c)(x
-a) = 0, where a, b, c R are always real. Find the
condition that the roots may be equal. What are the roots when this condition
is satisfied?

[**Hint.**= 2 ((a -b)² +(b -c)² +(c -a)²)] - Discuss the nature of roots of the following equations:

(i) 3 x² -2 x -3 = 0 (ii) x² -(p +1) x +p = 0

(iii) (x -a)(x -b) = a b.

It is given that p Q, and a, b R. - Find m so that roots of the equation (4 +m) x² +(m +1) x +1 = 0 may be equal.
- Show that the roots of the equation x² +2 (3 a +5) x +2 (9 a² +25) = 0 are complex unless a = 5/3.
- If a, b, c, d R show that the roots of the equation (a² +c²) x² +2 (a b +c d) x + (b² +d²) = 0 cannot be real unless they are equal.
- Determine a positive real value of k such that both the equations x² +k x +64 = 0 and x² -8 x +k = 0 may have real roots.

## Answers

**1.**(i) pair of complex conjugates (ii) rational and unequal

(iii) real and unequal (iv) pair of complex conjugates

**2.**Roots are real and equal when a = b = 0

**3.**Roots are equal when a = b = c. Then roots are a, a

**4.**(i) real and distinct

(ii) rational and distinct when p 1; when p = 1, roots are rational and equal

(iii) real and distinct when a +b 0; when a +b = 0, roots are both 0.

**5.**5, -3

**8.**k = 16