Completion of Squares
Clement Radcliffe,Contributor

In this week's lesson we will continue to review quadratic, using both the formula and the factorisation methods. Let us begin with the solution to the homework given last week.

• Solve 2x² - 6x - 1 = 0

Using the formula x = ( (-b (+ or -) square root of b2 - 4ac)/2a)

From the equation, a = 2, b = -6 and c = -1

... x = ( (-6 (+ or -) square root of (-6)² - 4 x 2 x -1)/2 x 2)

... x = 6 (+ or -) square root of (36 + 8)/4)

x = 6 (+ or -) square root of 44/4

... x = 6 (+ or -) 6.63/4

... x = (6 + 6.63)/4 = 12.63/4 = 3.16

And x = (6 - 6.63)/4 = -0.63/4 = -0.16

Answer: x = 3.16 and -0.16

• Solve 2x² - 3x - 4 = 2 - 4x

First make the right side equal to zero:

... 2x² - 3x + 4x - 4 - 2 = 0

... 2x² + x - 6 = 0 Factorising

(2x - 3)(x + 2) = 0

... 2x - 3 = 0, that is x = 3/2

And x + 2 = 0, that is x = -2.

Answer: x = 3/2 and -2

Let us now proceed to use the COMPLETION OF SQUARES method to reorganize a quadratic expression.

COMPLETION OF SQUARES

The quadratic expression is changed to a format which enables you to determine the following:

(1) The axis of symmetry
(2) The maximum or minimum value

Given the quadratic expression x² + bx + c, the aim is to convert the expression to the form (x + d)² + k, where d and k are constants. The method requires two steps as follows.

• You are, therefore, required to CONVERT x² + bx to a PERFECT SQUARE of the form (x + d)^{2. This is based on the following equation:} (x + d)² = x² + 2dx + d²
  Given x² + 2dx, then d², the square of half the coefficient of x, must be added to complete the perfect square.
• When b² is added to make x² + bx a perfect square, then b² is also subtracted to avoid change in value of the expression. It follows, therefore, that x² + bx + c is converted to x² + bx + d² + c - d².

EXAMPLE

Express x² + 6x + 3 in the form (x + d)² + k, where d and k constants.
Convert x² + 6x + 3 to the form (x² + 6x) + 3. As shown above 9 is added to x² + 6x to make it perfect square. When 9 is added then 9 is also subtracted to avoid changinng the value of the expression.

... (x² + 6x) + 3 = (x² + 6x) + (6/2)² + 3 - (6/2)²

= (x² + 6x + 9) + 3 - 9
= (x + 3)² - 6

... d = 3 and k = -6

(x + d)² + k, where d and k are constants. The method requires two steps as follows.

• You are therefore, required to CONVERT x² + bx to a PERFECT SQUARE of the form (x + d)². This is based on the following equation:
  (x + d)² = x² + 2dx + d²
  Given x² + 2dx, then d², the square of half the coefficient of x, must be added to complete the perfect square.
• When b2 is added to make x² + bx a perfect square, then b² is also subtracted to avoid change in value of the expression, It follows, therefore, that x² + bx + c is converted to x² + bx + d² + c - d²

EXAMPLE

Express x² + 6x + 3 in the form (x + d)² + k, where d and k are constants.
Convert x² + 6x + 3 to the form (x² + 6x) + 3. As shown above 9 is added to x² + 6x to make it perfect square. When 9 is added then 9 is also subtracted to avoid changing the value of the expression.

... (x² + 6x) + 3 = (x² + 6x) + (6/2)² + 3/2 - (6/2)²

= (x² + 6x + 9) + 3 - 9

= (x + 3)² - 6

... d = 3 and k = -6

EXAMPLE

Express 2x² - 3x + 1 in the form a(x + h)2 + k where a, h and k are real numbers.
2x² - 3x + 1 = 2(x² - (3/3)x) + 1

Complete the squares inside the bracket.

... 2(x² + (3/2)x + (3/4)² + 1

NB (3/4)² is added within the bracket so the real value is 2 (3/4)²2

= 2 (x² - (3/2)x + 9/16) - 9/8 + 1

= 2 (x - 3/4)² - 1/8

You may expand to show that the value of the expression remain the same.

... a = 2, h = -3/4 and k = -17/8

We will complete the lesson with another example.

EXAMPLE

Express f(x) = 2x² - 4x - 13 in the form f(x) = a (x + h)² + k

= 2x² - 4x - 13 = (2x² - 4x) - 13

= 2 (x² - 2x) - 13

Completing the square

... = 2(x² - 2x + 1) - 2 x 1 - 13
= 2(x - 1)² - 15
... f(x) = 2(x - 1)² - 15

For Homework please attempt the following:

1. Solve the equation x² + 4x - 3
2. Solve the equation 2x² + 5x = 9
3. Solve the equation 4x² + 9x + 10 = 4 - 2x
4. Express 2x² + 4x - 7 in the form a(x + b)² + c
5. Express 3x² - 2x + 1 in the form a(x + b)² + c

Clement Radcliffe is principal of Glenmuir High School. Send questions and comments to kerry-ann.hepburn@gleanerjm.com