|
Algebra
Clement Radcliffe,Contributor
We
will continue with the review of algebra.
Let us solve together the following
quadratic equations. Please be reminded
that you must be proficient in factorising
quadratic factors in order to be able
to master this method.
Solve
the following:
x2
- 3x- 4 = 0. Factorising the left-hand
side
x2
- 3x- 4 = (x - 4)(x +1)
:.(x
- 4)(x + 1) = 0
:.x
- 4 = 0, that is, x = 4 OR x + 1 =
0 , that is, x = - 1
Answer:
x = 4 and - 1
6x2
- x -15 = 0
:.(2x
+ 3)(3x - 5) = 0
:.
2x + 3 = 0, that is, x = -3⁄2
OR 3x - 5 = 0 , that is, x = 5⁄3.
Answer:
x = -3x⁄2
and 5x⁄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
Solve
the equations:
3a
- 1/2b = 4 (1)
9a
+ 2b = -2 (2)
Multiply
equation (1) by 4
12a
- 2b = 16 (3)
Add
equations (2) and (3)
21a
= 14
a
= 14⁄21
= 2⁄3
Substitute
into equation (2)
9
x 2⁄3
+ 2b = - 2
6
+ 2b = - 2
2b
= - 8
b
= - 4
Ans
: a = 2⁄3
and b = - 4
Most
quadratic equations cannot be solved
by factorisation. Alternatively, the
formulated 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 ±?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 ± vb2
- 4ac )/2a , then substituting
:.x
= ( - (-3 )± v(-3)2 - 4 x 2
x (- 7) )/2
x 2
:.x
= ( 3 ± v9 + 56 )/4
=
( 3 ±v65 )/4 = ( 3 ±
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 ± vb2
- 4ac )/2a
Substituting
the values above,
:.
x = ( 1± v1 - 4 x 2 x -2 )/4
= ( 1± v1 +16 )/4
:.x
= ( 1± v17 )/4 = ( 1±
4.12 )/4
:.x
= ( 1 + 4.12 )/4 = ( 5.12 = 1. 28
)/4
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 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
-b ± ( vb2 - 4ac )/2a 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.
Please
be sure to reflect the above in your
work in this topic in the future.
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 an independent contributor.
Send questions and comments to kerry-ann.hepburn@gleanerjm.com
|
