To find the nature of the roots of a quadratic equation, we use discriminant. The discriminant of the quadratic equation and denoted by the symbol Δ.
Formula to find discriminant Δ = b2 - 4ac.
Here,
a = coefficient of x2
b = coefficient of x
c = constant term
Value of Discriminant |
Nature of Roots |
Δ > 0 Δ = 0 Δ < 0 |
Roots are real an different Roots are real and equal Roots are imaginary |
Examine the nature of the roots of the following quadratic equations.
Example 1 :
x2 + 5x + 6 = 0
Solution :
Comparing
ax2 + bx + c = 0
and
x2 + 5x + 6 = 0,
we get a = 1, b = 5 and c = 6.
Find the value of the discriminant
Δ = b2 - 4ac
Δ = 52 - 4(1)(6)
Δ = 25 - 24
Δ = 1 > 0
Hence, the roots are real and unequal.
Example 2 :
2x2 - 3x + 1 = 0
Solution :
Comparing
ax2 + bx + c = 0
and
2x2 - 3x + 1 = 0,
we get a = 2, b = -3 and c = 1.
Find the value of the discriminant.
Δ = b2 - 4ac
Δ = (-3)2 - 4(2)(-1)
Δ = 9 + 8
Δ = 17 > 0
Hence, the roots are real and unequal.
Example 3 :
x2 - 16x + 64 = 0
Solution :
Comparing
ax2 + bx + c = 0
and
x2 - 16x + 64 = 0,
we get a = 1, b = -16 and c = 64.
Find the value of the discriminant
Δ = b2 - 4ac
Δ = (-16)2 - 4(1)(64)
Δ = 256 - 256
Δ = 0
Hence, the roots are real and equal.
Example 4 :
3x2 + 5x + 8 = 0
Solution :
Comparing
ax2 + bx + c = 0
and
3x2 + 5x + 8 = 0,
we get a = 3, b = 5 and c = 8.
Find the value of the discriminant
Δ = b2 - 4ac
Δ = 52 - 4(3)(8)
Δ = 25- 96
Δ = -71 < 0
Hence, the roots are imaginary.
Example 5 :
If the roots of the equation 2x2 + 8x - m3 = 0 are equal , then find the value of m.
Solution :
Comparing
ax2 + bx + c = 0
and
2x2 + 8x - m3 = 0,
we get a = 2, b = 8 and c = -m3.
Since the roots are equal, we have Δ = 0
x2 - 4ac = 0
82 - 4(2)(-m3) = 0
64 + 8m3 = 0
8m3 = -64
m3 = -8
m3 = (-2)3
m = -2
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