**Nature of Roots of Quadratic Equation Discriminant Examples :**

The roots of the quadratic equation ax^{2} +bx +c = 0 , a ≠ 0 are found using the formula x = [-b ± √(b^{2} - 4ac)]/2a

Here, b^{2} - 4ac called as the discriminant (which is denoted by D ) of the quadratic equation, decides the nature of roots as follows

Value of discriminant Δ = b Δ > 0 Δ = 0 Δ < 0 |
Nature of roots Real and unequal roots Real and equal roots No real roots |

**Example 1 :**

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them :

(i) 2 x² - 3 x + 5 = 0

Discriminant = b² - 4 a c

a = 2 b = -3 and c = 5

= (-3)² - 4 (2) (5)

= 9 - 40

= - 31 < 0

It has no real roots.

(ii) 3 x² - 4 √3 x + 4 = 0

**Solution :**

Discriminant = b² - 4 a c

a = 3 b = - 4 √3 and c = 4

= (- 4 √3)² - 4 (3) (4)

= 16 (3) - 48

= 48 - 48

= 0

It has equal real roots.

3x² - 2√3 x - 2 √3 x + 4 = 0

3x (√3 x - 2) + 2 (√3 x - 2) = 0

(3 x + 2) (√3 x - 2) = 0

3 x + 2 = 0 √3 x - 2 = 0

3 x = -2 √3 x = 2

x = -2/3 x = 2/√3

(iii) 2 x² - 6 x + 3 = 0

**Solution :**

Discriminant = b² - 4 a c

a = 2 b = - 6 and c = 3

= (-6)² - 4 (2) (3)

= 36 - 24

= 12 > 0

It has two distinct real roots.

We cannot factorize the given equation. To solve this we have to use the quadratic formula

x = (- b ± √ b² - 4 a c)/2a

x = [-(-6) ± √12]/2(2)

x = [6 ± √12]/4

x = [6 ± 2√3]/4

x = 2 [3 ± √3]/4

x = (3 ± √3)/2

**Example 2 :**

Find the values of k for which of the following quadratic equations, so that they have two equal roots.

(i) 2 x² + k x + 3 = 0

**Solution :**

Since the equation has two equal roots

Discriminant = 0

b² - 4 a c = 0

a = 2 b = k and c = 3

k² - 4 (2) (3) = 0

k² - 24 = 0

k² = 24

k = √24

k = √2 ⋅ 2 ⋅ 2 ⋅ 3

k = 2√6

(ii) k x (x - 2) + 6 = 0

**Solution :**

k x² - 2 k x + 6 = 0

Since the equation has two equal roots

Discriminant = 0

b² - 4 a c = 0

a = k b = -2 k and c = 6

(-2 k)² - 4 (k) (6) = 0

4 k² - 24 k = 0

4 k (k - 6) = 0

4 k = 0 k - 6 = 0

k = 0 k = 6

After having gone through the stuff given above, we hope that the students would have understood, how to find the nature of roots of quadratic equation using discriminant.

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