System of Equations

Quickly solve this system of equations:

x - y = 5
x^2 - y^2 = 35

Objects

There are 50 distinct objects lying on a table. I can choose any of the objects, and any of the objects can choose me. How many ways are there to pick the objects?

Counting Divisors

If n is the perfect cube of a natural number and has 28 divisors, 2 of which are prime, how many divisors does the cube root of n have?
(Source: Art of Problem Solving: Introduction to Number Theory)

Three Difficult Problems

13. There were a total of 1,050,697 crimes in the US way back in 1997. Newark, NJ housed 0.56% of these crimes. In 1998, Newark City officials started a safety program to lower their crime rate. During the year of 1999, Newark was home to only 5,540 crimes. A program is considered successful if it reduces its…well, whatever it’s testing…by at least 5%. Was this program successful?

14. Jimmy, a novice salesperson, earns a base pay of $200 plus 5% commission on his weekly sales. His goal commission is $100. Right now, Jimmy totals $1200 in sales. How much does Jimmy need to raise his total sales each week in order to reach his goal in two weeks? (Example: if your answer was $200, he would make $200 more every week, leading to $1600 at the end)

15. Frank needs to raise $2000 in two years in order to pay off his mortgage. However, every year he pays $500 back to the government for taxes. At what percent does the interest need to be in order to achieve his goal? Assume he starts at $1000.

Functions Galore!

DO NOT USE A CALCULATOR!

1. Define a @ b as a @ b = a^2 - b. Calculate 3 @ (5 @ 6).

2. Define $(a,b,c) to be $(a,b,c) = a(b-c). Calculate $($(0,8,6),$(6,3,4),$(1,7,5)).

3. Graph y = -2sqrt(x-3)+1.

4. In a country, population is determined as a function of time. Officials produce population patterns by using the function P(t) = 0.5t^2 + 0.358t. Determine the population (to the nearest whole number) after 110 years of growth. (Hint: 110=100+10)

Pythagorean Theorem Extended

If a < b, then 3^2 + 4^2 + 5^2 + 12^2 = a^2 + b^2 is represented by FOUR ordered pairs (a,b). What is the area of the figure that is bounded by these four ordered pairs if they were graphed on a Descartes grid?

Solving Quadratics by Completing the Square

Some quadratics are easy to factor, such as x^2 + 4x + 4, which turns out to be (x+2)^2. But say you have the equation x^2 + 4x + 2 = 0. Obviously, you can not factor that. However, you can factor x^2 + 4x + 4, as shown above. So basically, we need to add 2 to both sides of the equation in order to make it factorable. This is what completing the square is. You take a non-factorable quadratic and try to add a value that will make it a perfect square trinomial (one that can be factored into a value squared, like the one above)

Why a perfect square trinomial, you ask? Because the zero-product property does not work in this case. The factors could be anything. If the trinomial is a perfect square, then there are only two options for the number: it and its opposite. Therefore, the expression on the right side has a +- (plus or minus) after the square root is taken.

Let's work on that original problem. We have x^2 + 4x + 2 = 0. Now, we need to make the left side a perfect square. To do this, we add 2 to both sides, as mentioned. This results in x^2 + 4x + 4 = 2. Now, we factor and simplify.

(x+2)^2 = 2
 x+2    = +-sqrt(2)         
 x      = -2 +- sqrt(2)     

However, this will not work for a-coefficients greater than 1 (except for the perfect squares). We will have to divide the trinomial by the a-coefficient to set it up.

For an example, take 2x^2 + 12x + 7 = 0. The completing the square method is then as follows.

2x^2 + 12x + 7  = 0
x^2  + 6x + 3.5 = 0     Divide by 2
x^2  + 6x + 9   = 5.5   Perfect square is x^2 + 6x + 9
(x + 3)^2       = 5.5
 x + 3          = +-sqrt(5.5)
 x              = -3 +- sqrt(5.5)

However, if we were given 4x^2 + 8x + 1 = 0, we could just add 1 and simplify to get (2x + 2)^2 = 1. (Note: Since the answer here is a whole number, the trinomial is, indeed, factorable)


Some extra points:

• The completing the square method will always work. However, if it gets really messy, just use the quadratic formula.


• You do not need to guess on the number to add. Once you set up the equation in the form x^2 + bx = c, the number to add is (b/2)^2. This is based off the middle term and the last term of the (a+b)^2 expansion.