Nature of roots of a Quadratic Equation

Discriminant

      = b² -4ac

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

1. If = b² -4 a c = 0, then roots are equal (and real).
2. If = b² -4 a c > 0, then roots are real and unequal.
3. 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:

1. If = b² -4 a c = 0, then roots are rational and equal.
2. If = b² -4 a c > 0, and is a perfect square of a rational number, then roots are rational and unequal.
3. 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.
4. 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

1. 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.
2. 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.
3. 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.
4. 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)

1. 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)
2. Roots are real and unequal when
     = (m +6)(m -2) > 0 i.e. when m < -6 or when m > 2
3. Roots are a pair of complex conjugates when
     = (m +6)(m -2) < 0 i.e. when -6 < m < 2

Exercise

1. 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
2. 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].
3. 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)]
4. 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.
5. Find m so that roots of the equation (4 +m) x² +(m +1) x +1 = 0 may be equal.
6. Show that the roots of the equation x² +2 (3 a +5) x +2 (9 a² +25) = 0 are complex unless a = 5/3.
7. 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.
8. 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