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