USING DISCRIMINANT

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

ax+ 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

ax+ 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

ax+ 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

ax+ 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

ax+ 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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