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
Discriminant Δ = b² - 4ac |
Nature of roots |
Δ > 0 Δ = 0 Δ < 0 |
Roots are unreal Roots are real and equal Roots are imaginary |
Example 1 :
Examine the nature of the roots of the following quadratic equation.
x² + 5x + 6 = 0
Solution :
If x² + 5x + 6 = 0 is compared to the general form ax² + bx + c = 0,
we get a = 1, b = 5 and c = 6.
Now, let us 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 :
Examine the nature of the roots of the following quadratic equation.
2x² - 3x + 1 = 0
Solution :
If 2x² - 3x + 1 = 0 is compared to the general form ax² + bx + c = 0,
we get a = 2, b = -3 and c = 1.
Now, let us 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 :
Examine the nature of the roots of the following quadratic equation.
x² - 16x + 64 = 0
Solution :
If x² - 16x + 64 = 0 is compared to the general form ax² + bx + c = 0,
we get a = 1, b = -16 and c = 64.
Now, let us 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 :
Examine the nature of the roots of the following quadratic equation.
3x² + 5x + 8 = 0
Solution :
If 3x² + 5x + 8 = 0 is compared to the general form ax² + bx + c = 0,
we get a = 3, b = 5 and c = 8.
Now, let us 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 :
If 2x² + 8x - m³ = 0 is compared to the general form ax² + bx + c =0,
we get a = 2, b = 8 and c = -m³.
Since the roots are equal, we have Δ = 0
b² - 4ac = 0
8² - 4(2)(-m³) = 0
64 + 8m³ = 0
8m³ = -64
m³ = -8
m³ = (-2)³
m = - 2
Hence, the value of "m" is "-2".
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