Thinking and Application Practice
MCR3U
Rational Expressions Thinking and Application Practice
1. Factor completely.
2
a)  x  2   3  x  2   2
b)  x  4   1
4
2. Evaluate the following without a calculator: 10000 2  9999 2 .
3. Determine all integer values of x so that y will always be an integer in the relations.
a) y  3 
4. Simplify:
5
x2
b) y  2 
6
x3
3
x 1
2
2
x 1
3
5. Graph the function: y 
 x  6  2x 2  x  6 .
x 2  4 x  12
6. A packaging company makes boxes with no tops. One style of box is made from cardboard 20 cm long and 10
cm wide. Equal squares are cut from the corners and the sides are folded up.
a) Express the volume of the box as a function of x.
b) Express the surface area of the open-topped box as a function of x.
c) Write a simplified expresssion for the ratio of the volume of the box to its surface area.
7. In an electrical circuit, two resistances x ohms and y ohms are connected in parallel. The total resistance, z
ohms, is given by the formula 1  1  1 .
z
x
y
a) Solve the formula for z.
b) Two resistances in parallel are 10 ohms and 12 ohms. What is the total
resistance?
8. The arithmetic mean of two positive numbers is one-half the sum of the numbers.
57
Eg. The arithmetic mean of 5 and 7 is
 6.
2
The harmonic mean of two numbers is the reciprocal of the mean of their reciprocals.
 5 1  7 1 
Eg. The harmonic mean of 5 and 7 is 

2


a) Show that the harmonic mean of 5 and 7 is
1

35
, which is less than the arithmetic mean.
6
35
.
6
b) Show that the harmonic mean of two distinct positive numbers, a  x and a  x , is always less than
the arithmetic mean.
9. Evaluate and state your answer as a fraction in lowest terms.
4
3
2
5
3
1
4
Quadratic Thinking Practice
1)
2)
3)
4)
5)
6)
Show that it is impossible to form a 20 𝑐𝑚 length of wire into a rectangle with area 30 𝑐𝑚2 .
For 𝑎, 𝑏, and ℎ real numbers, show that the roots of (𝑥 − 𝑎)(𝑥 − 𝑏) = ℎ2 are always real.
𝑎+𝑏
Prove that the minimum value for (𝑥 − 𝑎)2 + (𝑥 − 𝑏)2 occurs when =
. What is the
2
minimum value?
If 𝑓(𝑥) is a quadratic function such that (0) = 2, 𝑓(1) = 4, and 𝑓(2) = 16, determine 𝑓(3).
(𝑦 = 5𝑥 2 − 3𝑥 + 2)
Determine the equation of the tangent to the circle 𝑥 2 + 𝑦 2 = 25 at the point (4, 3).
(4𝑥 + 3𝑦 − 25 = 0)
Write an equation in standard form for the quadratic function whose graph passes through
1
(8, 0), (0, 8) and (−2, 0).
(𝑦 = 2 𝑥 2 + 3𝑥 + 8)