Page 3 - LN

2
Example 1 : Find the discriminant of the quadratic equation 2x - 4x + 3 = 0 .
Ans: Here a = 2, b = -4, c = 3

2
D = b - 4ac
2
= (- 4) - 4(2)(3)
= 16 - 24
= - 8
2
Example 2 : What is the nature of roots of the quadratic equation 4x - 12x - 9 = 0?
2
Ans: The given equation is of the form ax + bx + c = 0,
where a = 4, b = -12, c = -9.
2
2
D = b - 4ac = (-12) - 4(4)(-9)
= 144 + 144 = 288 > 0.
So, the given equation has real and unequal or distinct roots.

2
Example 3 : Find the value of k such that the quadratic equation 9x - 6x + k = 0 has
the real roots:

Ans : Here a = 9, b = -6, c = k
2
D = b - 4ac
2
= (- 6) - 4(9)(k)
= 36 - 36k
For real roots D 0
36 - 36k 0
36 36k
1 k

2
Example 4: What is the nature of the roots of the quadratic equation 2x - 4x + 3 = 0?
2
Ans: The given equation is of the form ax + bx + c = 0, where a = 2, b = -4, c = 3
2
D = b - 4ac
2
= (- 4) - 4(2)(3)
= 16 - 24
= - 820
So, the given equation has no real roots.

2
Example 5: Show that the equation x + ax - 4 = 0 has real and distinct roots for all real
values of a.
2
Ans: The given equation is x + ax - 4 = 0
2
D = b - 4ac
2
= (a) - 4(1)(-4)
2
= a + 16
2
Clearly D = a + 16 > 0 for all a R. Hence, the equation has real and distinct roots for
all real values of a.
3