Find the discriminant of the following quadratic equations and hence find the nature of roots :

(i) 3x² - 5x - 2 = 0

(ii) 2x² - 3x + 5 = 0

(iii) 16x² - 40x + 25 = 0

(iv) 2x² + 15x + 30 = 0

(i) The given equation is 3x² - 5x - 2 = 0.

Comparing it with ax² + bx + c = 0, we get
a = 3 , b = -5 , c = -2

∴ Discriminant = b² - 4ac

Putting values of a, b, c in formula

$(-5)^2 - 4 \times 3 \times -2 \\[0.5em] = 25 + 24 \\[0.5em] = 49 \gt 0$

Discriminant = 49; Hence, the given equation has two distinct real roots.

(ii) The given equation is 2x² - 3x + 5 = 0.

Comparing it with ax² + bx + c = 0, we get
a = 2 , b = -3 , c = 5
∴ Discriminant = b² - 4ac
Putting values of a, b, c in formula

$(-3)^2 - 4 \times 2 \times 5 \\[0.5em] = 9 - 40 \\[0.5em] = -31 \lt 0$

Discriminant = -31; Hence, the given equation has no real roots.

(iii) The given equation is 16x² - 40x + 25 = 0.
Comparing it with ax² + bx + c = 0, we get
a = 16 , b = -40 , c = 25
∴ Discriminant = b² - 4ac
Putting values of a, b, c in formula

$(-40)^2 - 4 \times 16 \times 25 \\[0.5em] = 1600 - 1600 \\[0.5em] = 0$

Discriminant = 0; Hence, the given equation has two equal real roots.

(iv) The given equation is 2x² + 15x + 30 = 0.
Comparing it with ax² + bx + c = 0, we get
a = 2 , b = 15 , c = 30
∴ Discriminant = b² - 4ac
Putting values of a, b, c in formula

$(15)^2 - 4 \times 2 \times 30 \\[0.5em] = 225 - 240 \\[0.5em] = -15 \lt 0$

Discriminant = -15; Hence, the given equation has no real roots.