Completing the Square Formula
Completing the square formula is used to do the analyzing at which point the quadratic expression has minimum/maximum value. It can be used for graphing a quadratic function. Completing the square formula can also be used for solving a quadratic equation and deriving the quadratic formula.
What is the Completing the Square Formula?
Completing the square formula is a technique or method to find the roots of the given quadratic equations. A quadratic equation in its standard form, in variable x, is ax^{2} + bx + c = 0, where a, b and c are any real numbers but a ≠ 0.
The formula for completing the square is :
\(ax^2 + bx + c \Rightarrow ( {x + m} )^2 + {\rm{ constant}}\)
where, m is any real number.

Example 1: Solve by completing the square formula: x^{2}10x+16=0
Solution:
The given quadratic equation is:
x^{2}10x+16=0
We will solve this by completing the square.
Here, the coefficient of x^{2} is already 1
The coefficient of x is 10.
The square of half of it is (5)^{2} =25
Adding and subtracting it on the lefthand side of the given equation after the x term:
x^{2 } 10x + 25  25 +16=0
(x5)^{2 } 25 + 16 = 0
[∵ x^{2 } 10x + 25 = (x 5)^{2} ]
(x  5)^{2} 9 = 0
(x  5)^{2} = 9
(x  5) = ±√9
Taking square root on both sides
x  5 = 3; x  5 =  3
x = 8; x = 2
Answer: x = 8, 2.

Example 2: What number should be added to x^{2 } 7x in order to make it a perfect square trinomial? Solve it by using completing the square formula.
Solution:
The given expression is x^{2 } 7x
Method 1:
The coefficient of x is 7
Half of this number is \(\dfrac{7}{2}\)
Finding the square,
\[\left(\dfrac{7}{2} \right)^2= \dfrac{49}{4}\]
Method 2:
Comparing the given expression with \(ax^2+bx+c\),
a = 1, b = 7
The term that should be added to make the given expression a perfect square trinomial is,
\[ \left( \dfrac{b}{2a}\right)^2 = \left( \dfrac{7}{2(1)} \right)^2= \dfrac{49}{4}\]
Thus, from both the methods, the term that should be added to make the given expression a perfect square trinomial is,
Answer: \(\dfrac{49}{4}\)