|
Review
of algebra
Clement Radcliffe, Contributor
 | World
Food Day 2006 national ceremony and exhibition at the Ardenne High School in Kingston
on Thursday, October 19, 2006. - Rudolph Brown/Chief Photographer |
I
do hope that you had an enjoyable Christmas holidays. It would have even been
more beneficial if you were able to find the time to review your mathematics lessons.
Remember the examination is less than FIVE months away. We
will continue with the review of ALGEBRA. Let us solve together the following
quadratic equations. Solve
the following: x²
- 9x +14 = 0. Factorising the left hand side x²
- 9x + 14 = (x - 2)(x - 7) ie.
(x - 2)(x - 7) = 0
ie.
x - 2 = 0, that is, x = 2 OR x = 7 = 0 , that is, x = 7 Answer:
x = 2 and 7 2x²
- x -15 = 0 ie.
(2x + 5)(x - 3) = 0
ie.
2x + 5 = 0, that is, x = - 5/2 OR x -
3 = 0/2 , that is, x = 3. Answer:
x = -5/2 and 3 x²
+ x = 6 ie.
x² + x
- 6 = 0
ie.
(x + 3)(x -2) = 0
ie.
x = -3 and 2. Solve:
y = 2x²
- 3x - 2 when y = 0. ie.
y = 2x2 -3x -2 = 0. Factorising
(2x
+ 1)(x-2) = 0
ie.
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
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 ie.
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
± vb²
- 4ac. | ,
then substituting | | | 2a | |
| ie.
x = | -(-3)
± v(-3)²
- 4x 2 x (-7) | | | 2
x 2 |
| ie
x = | 3
± 65 | = | 3
± 8.063 | | | 4 | | 4 |
| ie.
Either x = | 11.063 | OR
x | -5.063 | | | 4 | | 4 |
ie.
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:
| ie
x = | 1
± 1 -4 x 2 x -2 | = | 1
± 1 + 16 | | | 4 | | 4 |
| ie
x = | 1
± 17 | = | 1
± 4.12 | | | 4 | | 4 |
| ie
x = | 1
± 4.12 | = | 5.12 | =1.28 | | | 4 | | 4 | |
| And
x = | 1
- 4.12 | = | ñ3.12 | =-0.78 | | | 4 | | 4 | |
Answer
is x = 1.28 and -0.78 Unless
you are specifically directed, you should attempt to use 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 vb2 - 4ac over 2a
separating -b. In other words, the incorrect formula below is 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.
For
homework, please find the solution of the quadratic equations. (l)
x² + 3x
+ 1 = 0
(2)
2x² -
6x -1 = 0
(3) 7x²
+ 8x -2 = 10
(4) 2x²
-3x -4 = 2- 4x. Clement
Radcliffe is the principal of Glenmuir High School in May Pen. | 
