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### 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**.