Completion of squares
By Clement Radcliffe, Contributor

In this lesson, I attempted the review of the somewhat complex process of completion of squares. I will continue to do so by considering the following examples.

Examples
Express the equation
a) x² + 8x = 9
b) x² + 0x = 0
in the form (x+a)²=b

a) As the square of the half coefficient of x must be added to both sides, then given the equation x² + 8x = 9, the coefficient of x is 8.
...x² + 8x + (8/2)² = 9 + (8/2)²

Please be reminded that 16 must b added to both sides of the equation.
... x² + 8x + 16 = 9 + 16 = 25
... (x + 4)² = 25

b) Given the equation x² + 9x - 4 = 0
Then x² + 9x = 4
The value to be added to both sides is (9/2)²
...x² + 9x + (9/2)² = 4 + (9/2)² = 4 + 81/4
...(x + 9/2)² = 97/4

Please note that the above was covered adequately in last week's lesson. If you are comfortable with the exaples, et us proceed to work together the homework given last week.

1. Solve the equation x² + 8x - 9 = 0 using completion of squares.
From the example above, given the equation:
x² + 8x - 9 = 0, the x² + 8x = 9. This is of course, equivalent to (x + 4)² = 25.
...(x + 4)² = 25. Find the square root for both sides
...(x+4) = 5
...x+4=5 ...x=1
or x+4=-5 ... x=-9
Answer: x=1, or -9

2. Solve x² + 9x - 4 =0 using completion of squares.
From the example above also
x² + 9x = 4 is expressed in the form
...(x+9/2)² = 97/4 = 24.25 Taking the square root for both sides
...x + 4.5 = 4.92
... x + 4.5 = 4.92 ... x = 0.42
or x + 4.5 = -4.92 ... x = -9.42
Answer: x=0.42 0r -9.42

3) Solve x² = 7x + 5 using completion of squares.
Converting the equation to the usual form:
...x² = 7x = 5
As the coefficient of x is -7, the we add (-7/2)² to both sides.
...x² - 7x + (-7/2)² = 5 + (-7/2)² = 5 + 49/4
...(x - 7/2)² = 69/4 = 17.75 Taking the square root
...x - 3.5 = 4.15
... x = 3.5 + 4.15 = 7.65
OR x = 3.5 - 4.15 =-0.65
Answer: x=7.65 0r -0.65

Clearly it is in your best interest to review the lesson provided last Tuesday to complete the above, especially if you had difficulites.

Let us now proceed to review some selected past paper questions which involve the solution of equations.

* Solve simultaneously:
2x + 3y = 11 (1)
4x + 2y = 10 (2) (January 2000)

Multiply equation (1) by 2
4x + 6y = 22 (3)
Equation (3) minus (2).
... 4y = 12
... y = 3, Substituting into (1)
...2x + 9 = 11
...2x = 2 ...x=1
Answer: x = 1 and y = 3.

* Solve the equation: 3(x+2)² = 7(x+2). (January 1990)
Expanding 3(x² + 4x + 4) = 7x + 14
...3x² + 12x + 12 = 7x + 14
...3x² + 12x + 7x + 12 - 14 = 0
...3x² + 5x - 2 = 0
Factorizing: (3x - 1)(x + 2) = 0
...3x - 1 = 0
... x =
OR x + 2 = 0
...x = -2
Answer: x = , -2

Alternate solution
Given the equation 3(x+2)² = 7(x+2)
As (x+2) is common to all terms, then let x + 2 = t,
Substituting: 3t² = 7t
...3t² = 7t
...3t² - 7t = 0 Factorizing
t(3t - 7) = 0
...t = 0. OR
3t - 7 = 0 ...t=7/3

Since x + 2 = t
... x + 2 = 0 ... x = -2
OR x + 2 = 7/3
... x = 7/3 - 2 =
Answer: x = -2, .

* Solve the equation 3x² + 5x = 6, giving your answer correct to two decimal places. (June 1992)
Since 3x² + 5x = 6, giving your answers correct to two decimal places.
Since 3x² + 5x = 6
...3x² + 5x - 6 = 0

Using the Formula Method,

From the equation, a=3, b=5 and c=-6.
Substituting


* Express f(x) = 3x² - 12 + 5 in the form f(x) = a(x+b)² + c when a, b and c are constants.
Since f(x) = 3x² - 12x + 5
...f(x) = 3(x²-4x) + 5
From inspection, you are required to convert x² - 4x to a perfect square. This is done by adding 4.

...f(x) = 3 {(x² - 4x + 4)-4}+5
Note that the 4 is also subtracted to maintain the value of f(x)
...f(x) = 3(x² - 4x + 4) - 12 +5
= 3(x-2)² -7.
By inspection, a = 3, b = -2 and c=-7
You may check your answer by expanding the expression.
f(x) = 3(x-2)² - 7 = 3(x²-4x+4)-7=3x²-12x+12-7
...f(x) = 3x² - 12x + 5

Substituting in equation (1), y = 2x+3
When x = 0.91, then y = 2x - 3.31 + 3 + -3.62
Answer: x = 0.91, y = 4.82 or x = -3.31, y = -3.62

Next week, we review the solution of quadratic equations using the graphical method. I am therefore asking you to rview GRAPHS before our next lesson. You may practice by plotting the graph f(x) = 3x² - 2x -1

Clement Radcliffee is Principal of Glenmuir High School in Clarendon. Send your questions and comments to the CXC Study Guide, the Gleaner Company Ltd., 7 North Street, Kingston; or email us at jcampbell@gleanerjm.com.