Factorisation
By Clement Radcliffe, Contributor

HAPPY NEW Year! May your fondest dreams be realised. This, of course, should include your desire to be successful in your examinations.

The following are the answers to the last homework given.

* Solve the quadratic equation: x2 ­ 9x + 14 = 0.

Factorising x² - 9x + 14 = (x ­ 2)(x - 7)
... (x ­ 2)(x - 7) = 0
If x - 2 = 0, then x = 2
If x - 7 = 0, then x = 7
Answer: x = 2 and 7

* Solve: 2x²­ x -15 = 0
... (2x + 5)(x ­ 3) = 0

I am sure you appreciate the need to be comfortable with factorising quadratic expressions.

... If 2x + 5 = 0, then x = -5 ÷ 2
If x - 3 = 0, then x = 3
Answer: x = - 5 ÷ 2 and 3.

* Solve: 2x² - x -3 = 0
... (2x - 3)(x + 1) = 0
... x = 3 ÷ 2 and -1

* Solve: x² + x = 6
x² + x - 6 = 0
... (x + 3)(x - 2) = 0
... x = -3 and 2.

* Solve: y = 2x² - 3x - 2 when y = 0.
... y = 2x² - 3x - 2 = 0.
Factorising
(2x + 1)(x - 2) = 0
... x = - 0.5 and 2

Most quadratic equations cannot be solved by factorisation. Alternatively, the formula method is used.

Please be reminded that given the quadratic equation ax² + bx + c = 0, where a, b and c are constants, then it can be shown that
x =

This is the basis of the formula method as x is found by substituting the values of a, b and c into the formula.

Example:

* Express 2x² = 3x + 1 in the form ax2 + bx + c = 0 and find the values of a, b and c.
Given that 2x² = 3x + 1. Then 2x2 -3x - 1 = 0.

By comparing this equation with the required form ax² + bx + c = 0
... a = 2, b = -3 and c = -1.

Please be careful not to omit the negative sign.

Answer: a = 2, b = -3 and c = -1.

* Solve 2x² - 3x - 1 = 0. Using the Formula method:

From the equation, a = 2, b = -3 and c = -1.
(Note that the zero must be on the right hand side).

Given the formula

Then substituting

... x = 1.78

Also,

... x = -0 .28

Answer is x = 1.78 and -0.28

Let us try another example.
Solve the following equation using the quadratic formula:

2x² + 2x - 8 = 3x - 6.
2x² + 2x - 8 = 3x - 6
2x² + 2x - 3x - 8 + 6 = 0
2x² - x - 2 = 0

Having expressed the equation into the appropriate form, then a = 2, b = -1 and c = -2.

Using the formula

Answer is x = 1.28 and -0.78

Unless you are specifically directed you should attempt using the factorisation method before the Formula method.

POINTS TO NOTE

* Care should always be taken in manipulating the negative signs, as this provides the greatest challenge in this method.

* The ® enables you to obtain two roots.

The entire numerator is over 2a. A common error is to use (check)b2 - 4ac over 2a, separating -b. In other words, the incorrect formula

sometimes used.

* The value within the square root should always be positive. When this is not so, it usually implies an error in calculation. Please check your working.

* If the value within the square root is negative, then the equation has no real roots.

Please find the solution of the quadratic equations for homework.

(1) x2 + 3x + 1
(2) 2x2 - 6x -1 = 0
(3) 6x2 + 11x = 10
(4) 2x2 - 3x - 4 = 2 - 4x

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.