Theory:

How quadratic formula comes?
Consider the quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\).
 
Let us find the roots of this equation by the method of completing the square.
 
Divide the equation by \(a\).
 
x2+bax+ca=0
 
Move the constant to the right side.
 
x2+bax=ca
 
Add the square of one half of coefficient of \(x\) on both sides.
 
x2+bax+b2a2=ca+b2a2
 
x+b2a2=ca+b24a2
 
x+b2a2=b24ac4a2
 
Taking square root on both sides.
 
x+b2a=±b24ac4a2
 
x+b2a=±b24ac2a
 
x=b2a±b24ac2a
 
x=b±b24ac2a
 
Therefore, the roots of \(ax^2 + bx + c = 0\) are x=b+b24ac2a and x=bb24ac2a.
The formula for finding the roots of the quadratic equation \(ax^2 + bx + c = 0\) is:
 
x=b±b24ac2a
 
This formula is known as the quadratic formula.
Example:
1. Find the roots of \(2x^2 + 3x - 77 = 0\) by using quadratic formula.
 
Solution:
 
The given equation is \(2x^2 + 3x - 77 = 0\).
 
Here, \(a = 2\), \(b = 3\) and \(c = -77\).
 
Quadratic formula:
 
x=b±b24ac2a
 
Substitute the given values in the formula.
 
x=3±324×2×772×2
 
x=3±9+6164
 
x=3±6254
 
x=3±254
 
x=3+254 or x=3254
 
x=224 or x=284
 
\(x =\) 112 or \(x = -7\)
 
Therefore, the roots of the given equation are \(-7\) and 112.