### Theory:

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

Solution:

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:

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:

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.