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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.