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Theory:

Let us see the procedure to solve the quadratic equation by the method of completing the square.
Step 1: Write the given equation in standard form \(ax^2 + bx + c = 0\).
 
Step 2: Make sure the coefficient of \(x^2\) is \(a = 1\). If not, make it by dividing the equation by \(a\).
 
Step 3: Move the constant term to the right hand side of the equation.
 
Step 4: Add the square of one half of the coefficient of \(x\) to both sides. [That is, add \(\left(\frac{b}{2}\right)^2\).]
 
Step 5: Make the equation a complete square and simplify the right hand side.
 
Step 6: Solve for \(x\) by taking square root on both sides.
Example:
Find the root of \(2x^2 + 7x - 15 = 0\) by the method of completing the square.
 
Solution:
 
The given equation is \(2x^2 + 7x - 15 = 0\).
 
Here, the coefficient of \(x^2\) is \(2\). So, divide the equation by \(2\).
 
22x2+72x152=02
 
x2+72x152=0
 
Move the constant term to the right hand side of the equation.
 
x2+72x=152
 
Add the square of one half of the coefficient of \(x\) to both sides.
 
x2+72x+742=152+742
 
x2+2×74x+742=152+4916
 
Left hand side equation reminds the identity \((a + b)^2 = a^2 + 2ab + b^2\).
 
x+742=120+4916
 
x+742=16916
 
Taking square root on both sides.
 
x+74=±134
 
x=13474 or x=13474
 
x=64 or x=204
 
\(x =\) 32 or \(x = -5\)
 
Therefore, the roots of the given equation are 32 and \(-5\).
 
 
2. Find the roots of the equation \(x^2 + x + 2 = 0\) by completing the square method.
 
Solution:
 
The given equation is \(x^2 + x + 2 = 0\)
 
Here, the coefficient of \(x^2\) is \(1\).
 
\(x^2 + x + 2 = 0\)
 
Move the constant term to the right hand side of the equation.
 
\(x^2 + x = -2\)
 
Add the square of one half of the coefficient of \(x\) to both sides.
 
x2+x+122=2+122
 
x2+2×12x+122=2+14
 
x+122=74 \(< 0\)
 
The above equation cannot be possible because the square of any number cannot be negative.
 
Therefore, the given equation has no real roots.