Completing the Square Formula


Completing the square is converting a quadratic expression of the form ax+ bx +c to the vertex form  a(x+d)2 + e

Completing the square is used for:

  1. Converting a quadratic expression into vertex form
  2. Analyzing at which point the quadratic expression has minimum/maximum value
  3. Graphing a quadratic function
  4. Solving a quadratic equation
  5. Deriving the quadratic formula

What is the Completing the Square Formula?

To complete the square in the expression ax+ bx +c, first find:

 d =b/2a  and   e = c - b2/(4a) 

Substitute these values in:

ax+ bx +c = a(x+d)+ e

 

Solved Examples Using Completing the Square Formula

Example 1: 

Solve by completing the square.

x2-10x+16=0

Solution:  

The given quadratic equation is:

x2-10x+16=0

We will solve this by completing the square.

Here, the coefficient of x2 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 left-hand side of the given equation after the x term:

x- 10x + 25 - 25 +16=0

(x-5)- 25 + 16 = 0

[∵ x- 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- 7x in order to make it a perfect square trinomial?

Solution:       

The given expression is x- 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: \(\dfrac{49}{4}\)