Factorisation
ClementRadcliffe, Contributor

A happy New Year to you all! I do hope that you had an enjoyable Christmas holiday and that you used some of the time to review your mathematics lessons. Remember the examination is less than FIVE months away.

We will continue with the review of ALGEBRA, by solving together the following quadratic equations.

Solve the following:

x² - 9x +14 = 0. Factorisng the left hand side

x² - 9x + 14 = (x - 2)(x - 7)

... (x - 2)(x - 7) = 0

... x - 2 = 0, that is, x = 2 OR x - 7 = 0 ,

that is, x = 7

Answer: x = 2 and 7

2x² - x -15 = 0

...(2x + 5)(x - 3) = 0

... 2x + 5 = 0, that is, x = - 5/2 OR x - 3 = 0 , that

is, x = 3.

Answer: x = -5/2 and 3

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 = -1/2 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 = -b ± square root of b² - 4ac/2a

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

Examples:

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 = -l.

Please be careful not to omit the negative sign.

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

• Solve 2x² - 3x - 7 = 0. Using the Formula method:

From the equation, a = 2, b = -3 and c = -7.

(Note that the zero must be on the right hand side).

Given the formula: x = -b ± square root of b² - 4ac/2a , then substituting

:.x = - (-3 )± square root of (-3)² - 4 x 2 x (- 7)/2 x 2

:.x = 3 ± square root of 9 + 56/4

= 3 ± square root of 65 = 3/4 ± 8.063/4

:. Either x = 11.063/4 OR x = - 5.063/4

:.x = 2.766 OR -1.266

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: x = -b ± square root of b² - 4ac/2a

:. x = 1± square root of 1 - 4 x 2 x -2/4 = 1± square root of 1 +16/4

:.x =1± square root of 17/4 = 1± 4.12/4

:.x = 1 + 4.12/4 = 5.12/4 = 1. 28

And x = 1- 4.12/4 = -3.12/4 = -0.78

Answer is x = 1.28 and -0.78

Unless you are specifically directed, you should attempt to use the factorization method before the Formula method.

POINTS TO NOTE

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

á The ± enables you to obtain two roots.

á The entire numerator is over 2a. A common error is to use vb 2 - 4ac over 2a, separating

-b. In other words, the incorrect formula -b ± square root of b² - 4ac/2a is sometimes used.

á 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.

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

For Homework, please find the solution of the quadratic equations.

(l) x² - 4x - 8 = 0

(2) 2x² + 5x = 9

(3) p² - 2p -11= 0

(4) 2x² - 3x - 4 = 2- 4x.

Clement Radcliffe is the principal of Glenmuir High School in May Pen.