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.

Wind Power!

An airplane is riding in the wind. Halfway through, the plane increases speed by 15 m.p.h. The time takes three hours and the distance travelled is 500 miles. What was the original speed of the airplane?

Time yourself!

To start, time yourself to see how long it takes for you to calculate the answers. Post your time in the comments! And do not use a calculator!
(Note: #1 does not need to be written out in full.)

1. 2^100 - 2^99


2. 6! + 7!


3. 198/6


4. 678+789


5. 100^(-1/2)


6. 1234+2345+3456+...+6789


7. (2^15)(5^16)


8. 155^2 - 154^2

Welcome!

This is a blog designed for your mathematical needs! I will post problems as well as give "lectures", if you will. Have fun!