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Review
Clement Radcliffe,Contributor
This
week we will continue the review of
algebra with the solution to last
week's homework.
Points
to Note
The
following methods are used to solve
quadratic equations:
Factorisation
Formula
method
It
should be noted that the formula method
is used when the quadratic expression
cannot be factorised. Now let us review
the homework.
Homework
1.
Solve the equation x2 + 4x - 3 = 0
Using
the formula, x = ( -b ± vb2
- 4ac )/2a
From
the equation, a = 1, b = 4 and c =
- 3
Substituting,
x = ( -4 ± v42 - 4x1x-3 )/2x1
Simplifying,
x = ( - 4 ± v16 + 12 )/2 =
( - 4 ± v28 )/2
x
= ( - 4 ± 5.3 )/2
x
= ( - 4 + 5.3 )/2
x
= 1.3/2 = 0.65
x
= (- 4 - 5.3 )/2 = -9.3/2 = -4.65
Answer
x = 0.65 or -4.65
2.
Solve: 4x2 + 9x +10 = 4
- 2x
Since
4x2 + 9x + 10 = 4 - 2x
4x2
+ 9x + 2x + 10 - 4 = 0
4x2
+ 11x + 6 = 0
Factorizing:
( 4x + 3) (x + 2) = 0
4x
+ 3 = 0; 4x = -3 x = -3/4
x
+ 2 = 0 x = - 2
Answer
x = - 3/4 , or -2
3.
Express 2x2 + 4x -7 in
the form a(x + b)2 + c
2x2
+ 4x -7 is expressed as (2x2
+ 4x) - 7
(2x2
+ 4x) - 7 = 2(x2 + 2x)
- 7
Completing
the square within the bracket
2(x2
+ 2x) - 7 = 2(x2 + 2x +
1) - 2 - 7
=
2(x + 1)2 - 9
a
= 2, b = 1 and c = -9
4.
Express 3x2 - 2x + 1 in
the Form a(x + b)2 + c
(3x2
- 2x) + 1 = 3(x2 - 2/3
x) + 1
To
complete the square within the bracket
we add the square of half the coefficient
of x
3(x2
- 2/3 x) + 1 = 3(x2 - ?
x + ( 1/3 )2) - 3x(1/3)2
+ 1
=
3 (x - 1/3)2 - 3/9 + 1
=
3 (x - 1/3)2 + 2/3
a
= 3, b = -1/3, c = 2/3
Continuing,
we will review the application of
completion of squares.
These
are:
1.
Equation of the axis of symmetry.
2.
Maximum or minimum value of the expression.
Example:
Given
the expression (a) 2x2
+ 4x - 7
(b)
3x2 - 2x + 1 find the following
a)
The equation of the axis of symmetry
b)
The minimum value of the expression
Since
2x2 + 4x - 7 may be expressed
in the form
y
= 2(x + 1)2 - 9, then the
equation of the axis of symmetry is
given
as
x + 1 = 0, x = - 1
The
minimum value of y = -9
Do
you know why?
2(x
+ 1)2 is positive for all
real values of x. It follows that
for all
values
of x, the values of the expression
are all greater than - 9.
Since
3x2 - 2x + 1 may be expressed
in the form y = 3 (x - 1/3)2
+ 2/3
Then
the equation of the axis of symmetry
is x - 1/3 = 0; x = 1/3
The
minimum value of y = 2/3
Since
y = 3( x - 1/3)2 + 2/3
N.B.
When x = 1/3, then 3 (x - 1/3)2 =
0
y
= 2/3
For
all other values of x, 3 (x - 1/3)2
is a positive number
y
is greater than 2/3
2/3
is the minimum value
We
will continue the review of algebra
by returning to the solution of simultaneous
equations. This week I will deal specifically
with those cases in which one equation
is linear and one quadratic.
Simultaneous
equations - One linear and one quadratic.
The
substitution method is used.
Example:
Solve
the following equations:
y
= x2 + 3x - 7 . . . (1)
y
+ x = 5 . . . ........(2)
The
substitution method is used as follows:
From
equation (2), y = 5 - x
Substituting
y = 5 - x in equation (1),
5
- x = x2 + 3x - 7
x2
+ 3x + x - 7- 5 = 0.
x2
+ 4x - 12 = 0
Using
the factorisation method:
(x
+ 6)(x - 2) = 0
x
= 2 and - 6. Substituting in equation
(2),
y
= 3 and 11. Answers: x = 2, y = 3
and x = - 6, y = 11.
Kindly
note the following:
(a)
There are two sets of values because
of the quadratic equation.
(b)
The basic principles of algebra should
be well known, as they are required.
If
your solutions have large values,
for example 136, it is likely that
an error has been made. It is therefore
recommended that you check your working.
Here
is another example.
Example
Determine
two numbers whose sum is 9 and whose
product is 20, by solving a quadratic
equation.
Let
the numbers be x and y.
x
+ y = 9 . . . (1)
x
x y = 20 ... (2)
From
equation (1),
x
= 9 - y . . . (3)
Substituting
equation (3) in equation (2),
(9
- y) x y = 20
9y
- y2 = 20
y2
- 9y + 20 = 0 Factorising
(y
- 5)(y - 4) = 0
y
- 5 = 0. y = 5
OR
y
- 4 = 0 y = 4.
Substituting
into equation (1)
When
y = 5
5
+ x = 9 x = 4
When
y = 4 4 + x = 9 x = 5
Answer
y
= 5 and x = 4
OR
y
= 4 and x = 5.
Please
attempt to solve the following on
your own:
x22
+ 9y2 = 37
x
- 2y = -3
y
- x =1
y
= x2 - 3x + 4
Enjoy
the rest of the week.
Clement
Radcliffe is an independent contributor.
Send questions and comments to kerry-ann.hepburn@gleanerjm.com
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