Completing the square | Mathematics for Grade 12 PDF Download

Introduction

• Completing the square is a method in algebra that is used to write a quadratic expression in a way such that it contains the perfect square. In simple words, we can say that completing the square is a process where consider a quadratic equation of the ax² + bx + c = 0 and change it to write it in the form a(x + p)² + q = 0. This method is generally used to find the roots of a quadratic equation. 

Completing the Square Method

• The most common application of completing the square method is factorizing a quadratic equation, and henceforth finding the roots or zeros of a quadratic polynomial or a quadratic equation. We know that a quadratic equation of the form ax² + bx + c = 0 can be solved by the factorization method. But sometimes, factorizing the quadratic expression ax² + bx + c is complex or NOT possible. Let us have a look at the following example to understand this case.

For example:

• x² + 2x + 3 cannot be factorized as we cannot find two numbers whose sum is 2 and whose product is 3. In such cases, we write it in the form a(x + m)² + n by completing the square. Since we have (x + m) whole squared, we say that we have "completed the square" here. 

Completing the Square Formula

• Completing the square formula is a technique or method to convert a quadratic polynomial or equation into a perfect square with some additional constant. A quadratic expression in variable x: ax² + bx + c, where a, b and c are any real numbers but a ≠ 0, can be converted into a perfect square with some additional constant by using completing the square formula or technique.

Note: Completing the square formula is used to derive the quadratic formula.
Completing the square formula is a technique or method that can also be used to find the roots of the given quadratic equations, ax² + bx + c = 0, where a, b and c are any real numbers but a ≠ 0.

Formula for Completing the Square:

• The formula for completing the square is: ax² + bx + c ⇒ a(x + m)² + n
• where, m is any real number and n is a constant term.
• Instead of using the complex step-wise method for completing the square, we can use the following simple formula to complete the square. To complete the square in the expression ax² + bx + c, first find:
• m = b/2a and n = c - (b²/4a)
• Substitute these values in: ax² + bx + c = a(x + m)² + n. These formulas are derived geometrically. Let us study this in detail using illustrations in the following sections.

Completing the Square Formula Examples

Example 1: Using completing the square formula, find the number that should be added to x² - 7x in order to make it a perfect square trinomial.

The given expression is x² - 7x.
Method 1:

• Comparing the given expression with ax² + bx + c, a = 1; b = -7
  Using the formula, the term that should be added to make the given expression a perfect square trinomial is,
  (b/2a)² = (-7/2(1))² = 49/4.
  Thus, from both the methods, the term that should be added to make the given expression a perfect square trinomial is 49/4.

Method 2:

• The coefficient of x is -7. Half of this number is -7/2. Finding the square, (-7/2)² = 49/4

Example 2: Use completing the square formula to solve: x² - 4x - 8 = 0.

Method 1:

• Using formula, ax² + bx + c = a(x + m)² + n. Here, a = 1, b = -4, c = -8
  ⇒ m = b/2a = (-4)/2(1) = -2
  and, n = c - (b²/4a) = -8 - (-4)²/4(1) = -12
  ⇒ x² - 4x - 8 = (x - 2)² - 12.
  ⇒ (x - 2)² = 12
  ⇒ (x - 2) = ±√12
  ⇒ x - 2 = ± 2√3
  ⇒ x = 2 ± 2√3

Method 2:

• Let’s transpose the constant term to the other side of the equation: x² - 4x = 8. Take half of the coefficient of the x-term, which is -4, including the sign, which gives -2. Square -2 to get +4, and add this squared value to both sides of the equation:
  x² - 4x + 4 = 8 + 4
  ⇒ x² - 4x + 4 = 12
• This process creates a quadratic expression that is a perfect square on the left-hand side of the equation. Simply we can replace the quadratic with the squared-binomial form: (x - 2)² = 12
• Now, we've completed the expression to create a perfect-square binomial, let’s solve:
  (x - 2)² = 12
  ⇒ (x - 2) = ±√12
  ⇒ x - 2 = ± 2√3
  ⇒ x = 2 ± 2√3
  Using completing the square method, x = 2 ± 2√3.

Solving Quadratic Equations Using Completing the Square Method

• Let us complete the square in the expression ax² + bx + c using the square and rectangle in Geometry. Based on the method studied earlier, the coefficient of x² must be made '1' by taking 'a' as the common factor. We get, ax² + bx + c = a[x² + (b/a)x + (c/a)]. Now, we will consider the first two terms, x² and (b/a)x. Let us consider a square of side 'x' (whose area is x²). Let us also consider a rectangle of length (b/a) and breadth (x) (whose area is (b/a)x).
  
• Now, divide the rectangle into two equal parts. The length of each rectangle will be b/2a.
  
• Attach half of this rectangle to the right side of the square and the remaining half to the bottom of the square.
  

To complete a geometric square, there is some shortage which is a square of side b/2a. The square of area [(b/2a)²] should be added to x² + (b/a)x to complete the square. But, we cannot just add, we need to subtract it as well to retain the expression's value. Thus, to complete the square:
x² + (b/a) x = x² + (b/a)x + (b/2a)² - (b/2a)²
= x² + (b/a)x + (b/2a)² - b²/4a²
Multiplying and dividing (b/a)x with 2 gives, x² + (2⋅x⋅b/2a) + (b/2a)² - b²/4a²
By using the identity, x² + 2xy + y² = (x + y)²
The above equation can be written as,
x² + (b/a) x = (x + b/2a)² - (b²/4a²)
By substituting this in (1): ax² + bx + c = a((x + b/2a)² - b²/4a² + c/a) = a(x + b/2a)² - b²/4a + c = a(x + b/2a)² + (c - b²/4a)
This is of the form a(x + m)² + n, where,
m = b/2a
n = c - (b²/4a)

Example: We will complete the square in -4x² - 8x - 12 using this formula. Comparing this with ax² + bx + c, a = -4; b = -8; c = -12

Find the values of 'm' and 'n' using:
m = b/2a = -8/²(-4) = 1
n = c - (b²/4a) = -12 - (-8)²/4(-4) = -8
Substitute these values in: ax² + bx + c = a(x + m)² + n
We get: - 4x² - 8x - 12 = -4(x + 1)² - 8

Completing the Square Steps

To apply the method of completing the square, we will follow a certain set of steps. Given below is the process of completing the square stepwise:

• Step 1: Write the quadratic equation as x² + bx + c. (Coefficient of x² needs to be 1.)
• Step 2: Determine half of the coefficient of x.
• Step 3: Take the square of the number obtained in step 1.
• Step 4: Add and subtract the square obtained in step 2 to the x² term. 
• Step 5: Factorize the polynomial and apply the algebraic identity x² + 2xy + y² = (x + y)² to complete the square.

How to Apply Completing the Square Method?

Example: Complete the square in the expression -4x² - 8x - 12.

First, we should make sure that the coefficient of x² is '1'. If the coefficient of x² is NOT 1, we will place the number outside as a common factor. We will get:
-4x² - 8x - 12 = -4(x² + 2x + 3)

Now, the coefficient of x² is 1.

• Step 1: Find half of the coefficient of x. Here, the coefficient of 'x' is 2. Half of 2 is 1.
• Step 2: Find the square of the above number. 12 = 1
• Step 3: Add and subtract the above number after the x term in the expression whose coefficient of x² is 1. This means, -4(x² + 2x + 3) = -4(x² + 2x + 1 - 1 + 3)
• Step 4: Factorize the perfect square trinomial formed by the first 3 terms using the identity x² + 2xy + y² = (x + y)². In this case, x² + 2x + 1 = (x + 1)². The above expression from Step 3 becomes: -4(x² + 2x + 1 - 1 + 3) = -4((x + 1)² - 1 + 3)
• Step 5: Simplify the last two numbers. Here, -1 + 3 = 2. Thus, the above expression is: -4x² - 8x - 12 = -4(x + 1)² - 8. This is of the form a(x + m)² + n. Hence, we have completed the square. Thus, -4x² - 8x - 12 = -4(x + 1)² - 8

Note: To complete the square in an expression ax² + bx + c

• Make sure the coefficient of x² is 1.
• Add and subtract (b/2)² after the 'x' term and simplify.

Trick to Learn Completing the Square Method

Here are a few tips for completing the square technique.

• Step 1: Note down the form we wish to obtain after completing the square: a(x + m)² + n
• Step 2: After expanding, we get, ax² + 2amx + am² + n
• Step 3: Compare the given expression, say ax² + bx + c and find m and n as m = b/2a and n = c - (b²/4a).