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UNIT 3 : QUADRATIC FUNCTIONS & EQUATIONS
LESSON
9: CHAPTER TEST SOLUTIONS
1. Solve by factoring.
2. a)
Solve by completing the square:
2x2 + 8x – 5 = 0
b)
Without solving, determine the number of roots of the equation 2x2
– 3x +7 = 0.
Solution:
3. Solve using the quadratic formula where 
. Express answers in simplest radical form.
Solutions:
4. Find the maximum or minimum value of the
quadratic functions given below.
Solutions:
5. Simplify.
Solutions:
6. Simplify.
Solutions:
7. Last month Zack’s sold running shoes at $120
per pair. At this price they sold 90
pairs of shoes. A marketing research
survey shows that for each $10 price increase, sales will drop by 6 pairs per
month.
a) Determine the selling price that will
maximize revenue and state the maximum revenue at this price.
b) If the wholesale price of the shoes to Zack
is $50.00, find the selling price that will maximize profits.
Solution:
à The key step in these problems is the “Let” statement:
a) Let the number
of ($10.00) price increases be x.
Revenue (last month) = Quantity x
Price = 90 x 120
New
Price = 120 + 10x and new quantity = (90 – 6x).
New Revenue = Quantity x Price =
(90 – 6x)(120 + 10x)
8. A picture 20 cm wide by 10 cm high is to be
surrounded by a mat of uniform width.
If the area of the mat is to be twice the area of the picture, find the
width of the mat.
x
à Determine what
is unknown or what you are asked to find?
Assign variables to the unknowns. Here we are asked
to find the width of the lawn.
Let the
width of the lawn be x metres. See
diagram.
Therefore
the length of the larger rectangle will be (80 + 2x) and its width will be (60
+ 2x)
The area
of the smaller rectangle is 80 x 60 = 4800 m2. The area of the uniform lawn is 3200 m2.
Therefore
the area of the larger rectangle will be 4800 + 3200 = 8000 m2. The following equation results.
9. A farmer wants to fence a rectangular field
and then divide it in half by placing a fence down the middle. Find the dimensions of the field which would
maximize the area if he has 600 m of fence available.
y
x x x 
y
Solution:
à Determine
what is unknown or what you are asked to find?
Assign variables to the unknowns. Here the
dimensions of the field are unknown.
Let the length be y and the width be x.
à Determine
the quantity to be maximized or minimized.
We are asked to maximize the area. Hence our formula is:
A = xy ** This is your main function
for the problem
à Write this formula as a function of
one variable. Since the available fencing is 600
m, we have a secondary relation between the two variables.
Solution:
First
find the zeros or x – intercepts. Let y
= 0 and solve for x.
Next
find the vertex.
Now make
a table of values as above including points near the asymptotes x = -3 and x =
1
|
x |
|
|
|
-5 |
30 |
1/30 |
|
-3 |
8 |
1/8 |
|
-2.5 |
3.75 |
1/3.75 = 0.27 |
|
-2.1 |
0.71 |
1.41 |
|
-2.01 |
0.0701 |
14.26 |
|
-2 |
0 |
Undef.(VA) |
|
-1.99 |
-0.0699 |
-14.31 |
|
-1.9 |
-0.69 |
-1.45 |
|
-1.5 |
-3.25 |
-0.31 |
|
0 |
-10 |
-1/10 |
|
1.5 |
-12.25 |
-0.08 |
|
4 |
-6 |
-1/6 |
|
4.9 |
-0.69 |
-1.45 |
|
4.99 |
-0.0699 |
-14.31 |
|
5 |
0 |
Undef.(VA) |
|
5.1 |
0.71 |
1.41 |
|
5.01 |
0.0701 |
14.27 |
|
6 |
8 |
1/8 |
|
7 |
18 |
1/18 |
Note values of x taken near asymptotes:
x
= - 2.1, -2.01, - 1.9, -1.99, 4.9, 4.99, 5.01, 5.1
11. For what values of k does the
function f(x) = 4x2 – kx + 1 have two distinct zeros?
Solution:
For two
distinct zeros, b2 – 4ac
> 0
a = 4 b = -k c = 1
|
Interval |
|
|
|
|
Test Number |
- 5 |
0 |
5 |
|
Sign
of (x – 4) |
( - ) |
( - ) |
( + ) |
|
Sign
of (x + 4) |
( - ) |
( + ) |
( + ) |
|
Sign
of (x – 4)(x + 4) |
( + ) |
( - ) |
( + ) |
