Some example problems based on nature of roots — lesson. Mathematics State Board, Class 10.

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1. Find the discriminant of the quadratic equation

$2x^2 + 7x - 4 = 0$ , and hence find the nature of its roots.

Solution:

The given equation is

$2x^2 + 7x - 4 = 0$ .

Here,

$a = 2$ ,

$b = 7$ and

$c = -4$ .

Discriminant

$=$

$b^2 - 4ac$

$=$

$7^2 - 4(2)(-4)$

$=$

$49 + 32$

$=$

$81 > 0$

Thus, the equation has real and distinct roots.

2. Find the discriminant of the quadratic equation

$x^2 - 8x + 16 = 0$ , and hence find the nature of its roots.

Solution:

The given equation is

$x^2 - 8x + 16 = 0$ .

Here,

$a = 1$ ,

$b = -8$ and

$c = 16$ .

Discriminant

$=$

$b^2 - 4ac$

$=$

$(-8)^2 - 4(1)(16)$

$=$

$64 - 64$

$=$

$0$

Thus, the equation has real and equal roots.

3. Find the discriminant of the quadratic equation

$x^2 - 5x + 12 = 0$ , and hence find the nature of its roots.

Solution:

The given equation is

$x^2 - 5x + 12 = 0$ .

Here,

$a = 1$ ,

$b = -5$ and

$c = 12$ .

Discriminant

$=$

$b^2 - 4ac$

$=$

$(-5)^2 - 4(1)(12)$

$=$

$25 - 48$

$=$

$-23 < 0$

Thus, the equation has no real roots.

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2. Some example problems based on nature of roots

Theory: