The value(s) of y for which the following pair of equations
x2 + y2 – 16 = 0 and x2 – 3y + 12 = 0
may have a real, common solution are:
- 4 only
- -7, 4
- 0, 4
- no y
- all y.
Algebra I
Maximize Fraction, Bank Robber
You have just caught a bank robber and are about to collect a reward. The reward will be given to you according to this formula: Reward = 
and you can pick the values of x and y as follows:
x = -7, -2, 0, or 10;
y = -3, -2, 1, or 20.
y = -3, -2, 1, or 20.
Which values should you pick so that you maximize your reward?
Numerical Trinomial
Find two factors (other than 1 and the number itself) whose product is:
612 + 88 + 278
Factoring by Grouping
Factor completely: ac + a + bc – bd – ad + b.
Factoring Backwards (a – b)2
Find the value of this expression without using a calculator and without doing a lot of work. (Use your algebra eyes to see what’s going on.)
(1379)2 – (2) · (1379) · (379) + (379)2
Formula: Recursive to Closed
If f(1) = 2 and f(n) = f(n – 1) + n, then what’s a formula for f(n)?
Calendar Square of Dates: Formula
Take a calendar, and draw a square around any nine dates (it will be a 3 x 3 square, right?) Let s be the smallest number in the square. If you add up all the numbers in the square, the answer will always be a multiple of 9. In particular, the sum will be exactly 9 · (s + 8). Why?
Plywood, Circular Holes
A circular piece of plywood has a radius of x inches. Four holes, each having a radius of y inches, are removed from the piece. Write a formula for the area of the remaining wood. (These holes don’t overlap each other at all.) Evaluate your formula if x = 10 inches and y = 2 inches. (Use 
≈ 3.14).
Rent-a-Heap: Numbers & Letters
Harry’s Rent-a-Heap will rent you a rusted 1968 Ford van for $15 per day plus
$ .20 per mile.
- How much will you pay if you rent the van for 3 days and drive it 245 miles?
- How much will you pay if you rent it for d days and drive it m miles? (The answer is a formula.)
Box Volume Formula
A rectangular box with no top is formed by cutting squares from the corners of an
8 by 15 sheet of cardboard and then folding up the sides. If the square cutouts are each x by x, then (a) what formula (in terms of x) represents the volume of the resulting box? (b) Is this a quadratic function? (c) What is the domain of x-values that are relevant to this problem?