Factorisation
By Clement Radcliffe, Contributor

I WILL begin today's lesson by giving the solution to the homework you got last time.

* Solve the simultaneous equations:
3x + 2y = 19 .....(1)
xy = 15 ............(2)
From Equation (1), 2y = 19-3x
... y=19-3x/2

Substituting in equation (2),


Substituting in equation (1),

Solve the simultaneous equations:
x² - y² = 24 .....(1)
y = 24 + 3 .......(2)

Substituting equation (2) into (1)
... x² = (2x + 3)² = 24
... x² + 4x² + 12x + 9 =24
... 5x2 + 12x +9 - 24 =0
... 5x² +12x - 15 = 0

From inspection, Factorization Method cannot be used. Using the Formula Method:

From the equation,


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

Solve the equations: x + y = 5
xy = 6
x + y = 5 ......(1)
xy = 6 ..........(2)
From equation (1), y = 5 - x
Do you notice that you could have found x just as easily?
Substituting into (2)
... (5-x)x = 6
... 5x -x² = 6, that is x² - 5x + 6 = 0
... (x - 3)(x - 2) = 0
... x = 3,2 Substituting into (1)
... y = 2,3
Answer: x = 3, y = 2 and x = 2, y = 3

Quadratic equations may also be solved using the method of Completion of Squares.

COMPLETION OF SQUARES

Given the equation x² + bx + c = 0, the aim is to convert the equation to the form (x+d)² = k, where d and k are constants. You are therefore required to CONVERT the left had side to a PERFECT SQUARE of form (x+d)². Given the form (x+d)² = k, then x is found by determining the square root of both sides.

Example:
Solve x² + 6x - 5 = 0, using completion of squares.
x² + 6x - 5 = 0
It is necessary to remove the constant to the right hand side:
x² + 6x = 5
Completing the squares on the left hand side:


POINTS TO NOTE (From the example above)
* The critical step is to transfer the constant to the Right Hand Side (R.H.S) and then make the Left Hand (L.H.S.) a perfect square. This is based on the following equation:
(x+a)² = x² + 2ax + a²
Given x² + 2ax, then a², the square of half the coefficient of x, must be added to complete the square.

Examples

Now let us solve the above examples together.
(a) Solve x² -8x = 1

If x² - 8x = 1
Completing the squares:
...x² - 8x + 16 = 1+16


(b) Solve x² - 12x =1
Since x² - 12x = 1
Completing the squares:
...x² - 12x + 36 = 1 + 36
... (x-6)2 = 37


Set out below is your homework.
Sove the following equations, using the completion of squares method.
(1) x² + 9x - 4 =0
(2) x² + 8x - 9 = 0
(3) x² = 7x + 5

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.