Completing the Square

In algebra, completing the square is a technique used to rewrite a quadratic expression in a form that is a perfect square. For example, a quadratic equation like:

   ax² + bx + c = 0 

It can be rewritten (by completing the square) in the form:
 
a(x + p)² + q = 0

Where p and q are numbers we calculate during the process.  
 

We can use the process of completing the square to find the roots or zeros of a quadratic equation or quadratic polynomial, and to factorize the equation. When the expression is not possible to factorize, this technique plays an important role.

For instance, x² + 2x + 3 cannot be factorized using real numbers, because there are no two numbers that add to 2 and multiply to 3. In such cases, we use completing the square to rewrite it in a new form: 
 
a (x + m)² + n   


This form helps us express the quadratic as a perfect square plus a constant.
   
Here, by rewriting the expression as (x + m), we complete the square. 
 

We have to follow certain steps for the method of completing the square. 
 

Step 1: Express the quadratic equation as x² + bx + c. 
Here, 1 must be the coefficient of x². If it is not 1, the number will be a common factor and placed outside. 

Step 2: Find the half of the coefficient of x.

Step 3: Find the square of the number (half of the coefficient). 

Step 4: Add and subtract the square within the expression to maintain equality.

Step 5: To complete the square, factorize the polynomial and use the algebraic identity.   

x² + 2xy + y² = (x + y)²

Or

x² - 2xy + y² = (x - y)²

Now, let us take an example for better understanding. 

Complete the square for -4x² - 8x - 12.
 

Step 1: First, we need to find if the coefficient is 1. Here, the coefficient of x² is not 1, so the number (-4) is placed outside as a common factor.  

-4x² - 8x - 12 = -4 (x² + 2x + 3)

However, the coefficient of x² is now 1. 

Step 2: Next, we need to find half of the coefficient of x. 
Coefficient of x = 2
Half of 2 = 1 

Step 3: Find the square of 1.
1² = 1 

Step 4: Add and subtract the square to the x² term. 
 -4 (x² + 2x + 3) = -4 (x² + 2x + 1 - 1 + 3) 

Step 5: Factorize the polynomial and apply the algebraic identity. 

x² + 2xy + y² = (x + y)²

Here, the first three terms: x² + 2x + 1 
 

• Last two terms: -1  + 3 
  
  x² + 2x + 1 = (x + 1)²
   -4 (x² + 2x + 1 - 1 + 3) = -4 ((x + 1)² - 1 + 3)
   

• Next, simplify the last two terms: -1 + 3
  -1 + 3 = 2
   

• Now, the expression becomes:
  -4 (( x + 1)² + 2)
   

• Next, we can distribute the -4:
  -4 (x + 1)² - 8
   
• Hence, the final result is:
  -4x² - 8x - 12 = -4 (x + 1)² - 8
   

• Now, this is the completed square form:
   a (x + m)² + n 

Converting a quadratic equation or polynomial into a perfect square with an extra constant is known as the process of completing the square. 

For a quadratic expression:  ax² + bx + c = 0

Here, a, b, and c are real numbers, but a is not equal to 0. 

The formula for completing the square is:

ax² + bx + c = a (x + m)² + n 

Here,

m = b / 2a 
n = c - (b² / 4a) 

When completing the square of a given expression, we first need to find the values of m and n, then substitute these values into the formula. 

Now, let us understand the application of the formula for completing the square. 

For example, x² - 4x - 8 = 0 

Now, we can apply the formula for completing the square:
ax² + bx + c = a (x + m)² + n
 

• Here,
  
  a = 1
  b = -4 
  c = -8 
   
• Now, we can find the value of m and n.
  
  m = b / 2a = -4 / 2 (1) = -2
  n = c - (b² / 4a) = -8 - ((-4)² / 4 (1)) 
  n = -8 - (16 / 4) = -8 - 4 
  n = -12 
   

• So, the expression becomes: 
   x² - 4x - 8 = (x - 2)² - 12 
   

• Now, we can solve the expression. 
   
   (x - 2)² - 12 = 0
     (x - 2)² = 12 
     x - 2 =  √12  =  2√3 
     x = 2 ∓ 2√3 
  
  Therefore, x = 2 + 2√3 or x = 2 - 2√3 

• For geometric visualization, divide the rectangle. So, b / 2a will be the length of each rectangle.
   

• Fix one rectangle to the right side of the square and the next one to the bottom of the square. 
   
• To complete the geometric square, we need to add a square of area [ (b / 2a)² ] to x² + (b / a) x. To retain the value of the expression, we must subtract it. Thus, to complete the square:
  
  x² + (b / a) x = x² + (b / a) x + (b / 2a)² - (b / 2a)²
  x² + (b / a) x = x² + (b / a) x + (b / 2 a)² - b² / 4a²
   

• Multiplying and dividing (b / a) x with 2 gives, 
  x² + (2x × b / 2a) + (b / 2a)² - b² / 4a²
   

• Now, we can use the identity, x² + 2xy + y² = (x + y)² 
  
  The equation above can be expressed as, 
        x² + (b / a) x = (x + b / 2a)² - (b² / 4a²) 
   

• Now, we had:
   ax² + bx + c = a (x² + b / ax) + c
   

• Next, substitute the completed square form:
   a ((x + b / 2a)² - b² / 4a²) + c
   

• Now, distribute the ‘a’: 
  a (x + b / 2a)² - a (b² / 4a²) + c 
   

• Then, simplify:
  a (b² / 4a²) = b² / 4a 
   

• Therefore, the expression becomes:
  a (x + b / 2a)² - b² / 4a + c
   

• Next, arrange the constants: 
  a (x + b / 2a)² - (c - b² / 4a)
   
• Thus, ax² + bx + c = a (x + b / 2a)² - (c - b² / 4a)
   

• The expression follows the form:
    a (x + m)² + n 
  
  Where, m = b / 2a and n = c - b² / 4a 

Understanding the significance of the method of completing the square is useful since it can be applied in various situations. Here are some real-world applications of the concept.
 

• In engineering and construction, engineers use the method of completing the square to ensure proper stability and balance. For example, if they need to construct a building that has different shapes, such as squares and rectangles, professionals can use the technique to find the dimensions. 
   
• Astronomers and space scientists can use the technique of completing the square to analyze the satellites and orbits of planets.  For instance, to analyze and predict the path of satellites, researchers use the completing the square method. 
   
• Companies and organizations can analyze their profit and break-even points using completing the square process. For example, if the profit is P(x) = -4x² + 24x - 10, completing squares can determine the number of items needed to maximize profits.