# USING DISCRIMINANT

## About "Using discriminant"

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 unrealRoots are real and equalRoots are imaginary

## Using discriminant - Examples

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". After having gone through the stuff given above, we hope that the students would have understood "Using discriminant".

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