Completing the Square and Related Craziness

Try these: Note: the notation ^2 will mean squared as I do not know how to create superscripts in html.

FOIL

1. (x+2)^2
2. (x-3)^2
3. (x-4)^2
4. (x+5)^2
5. (x+3/4)^2
6. (x+k)^2
7. (3x+2)^2
8. (3x-1)^2
9. (4x-4)^2
10. In a perfect square trinomial (the result of squaring a binomial) ______ of the coeffiecient middle term is always equal to the square root of the last term.
11. Another way of saying that is the _______ of ____ of the middle term is ______ to the last term.

Analyze:

Determine if the following are perfect square polynomials. If they are write them as the square of a binomial. If not show me why and write "not a perfect square".Here's an example of what I'm looking for:

Example: x^2 + 12x + 36

Possible Answers:

"Yes, because 12 divided by two is 6 and 6^2 = 36. The last term is the square of half the coefficient of the middle term."

or

"Yes, because the square root of 36 doubled is 12"

1. x^2 + 5x + 10
2. x^2 + 3x + 9
3. x^2 + 6x + 12
4. x^2 - 2x + 1
5. x^2 - 5x + 25
6. x^2 -3x + 9/4
7. x^2 - 8x + 16

Evaluate:

Find the value of c for which ax^2 + bx + c is a perfect square trinomial

Hint: c must always equal (b/2)^2 for ax^2 + bx + c to factor into (x + b/2)^2

1. x^2+ 6x + c
2. x^2 + 14x + c
3. x^2 - 8x + c
4. x^2 - 16x + c
5. x^2 + 5x + c
6. x^2 -2/5x + c

Solve:

Finally use the method of completing the square to solve the following problems

Example: 2x^2 -3x = 5

Explanation:

Divide everything by 2 to get x^2 -3/2x = 5/2
then divide -3/2 by 2 (that is -3/4) and add the square of -3/4 to both sides.
That is, add 9/16 to both sides.
Your result is x^2 -3/2x + 9/16 = 5/2 + 9 /16.
Now the left side can be factored to the square of a binomial and the right can be simplified into 41/16.
So x^2 - 3/2x + 9/16 becomes (x - 3/4)^2 = 41/16
This result can be solved by employing the squareroot property in the same fashion as you did for the ones above.

1. x^2 + 6x = 5
2. x^2 - 16x = -70
3. x^2 +18x = -6
4. y^2 + 20y = -40
5. m^2 + 18m + 100 = 0
6. 2x^2 - 16x = 6
7. 5t^2 - 60t = -20
8. 2x^2 +16x -8 = 0
9. m^2 + 8m -9 = -5

Hint: On the 6 - 8, you will need to divide by the
coeffecient of the x^2 term before doing anything else.

General outline of Final

You may print this without printing the whole blog by clicking this more printable link and then printing off the page it takes you to. Note: this outline is based off an old final. Additional concepts discussed from geometry and like solving for powers and bases in exponential functions by employing properties of natural logs and nth roots though not mentioned below will also be tested. Please review notes on how to solve those as well as stuff on quadratics and linear equations mentioned below. It is hard for me to put a precise outline until after everything is written and I'm always fiddling with my finals even the night before.

Chapter 1

1. question involving order of operation
2. solving equations with fractions in them
3. solving percent type problems _____ % of _____ is ______ or something similar
One can write an equation of use IS over OF = % over 100 technique discussed in class
4. Using proprtional reasoning to find an unknown quantity
5. Writing equations to solve problems with consecutive integers (even, odd, or otherwise) in them

Chapter 2

6. Graphing equations
7. What is an x or y-intercepts
8. Finding them.
9. Graphing equations, determining the slope and y-intercept
10. "
11. Finding the slope and y-intercept of a line passing through two points. Writing the equation.
12. "
13. Writing the equation of a line parallel or perpendicular to a given line.
14. Writing the equation of a line parallel or perpendicular to a given line passing through a given point.

Chapter 3

15. Writing a function for an applied problem (it will be a linear function so try to determine which the independant variable (x) is in the problem and which is the dependant variable (y) that is increasing or decreasing as a result. Use this to make a chart of (x,y) ordered pairs (you won't have many) from which you can figure out the slope and also the y-intercept to write the equation. Keep in mind that the y-intercept is always the initial value so you can back track in your table to find it if you must. Write the function in f(x) = mx + b form. You might have to make a graph of this function and/or use your function to predict future values. Kinda like pg. 82, lab exercise 4 or pg.106-107,ex 16. or pg. 109/17

That last one is worth a good amount. Be sure you know what your doing.

16. Finding a linear function from a table. Section 3-3/ex 16 and hw problems 9-12
17. Writing a simple linear cost function from a word problem. Like 3-3/ex 13 and 108/4-6

Be sure to copy that purple box on page 104. It will be helpful to know if you do not already.

18. Using number 17 equation to answer a question.

19. direct, inverse, or combined variation problem
20. problem like the rocket lab. know how to imput values of time into the function to find values for the height, H(t) at those specific times, H(0) = the initial height, the rocket hits the ground where H(t) = 0 (the time is found using the quadratic equation after the function is set = to zero), the projectile reaches its maximum at t = -b/2a, so that the max height is H(-b/2a), know how to determine when at what two times a projectile will be at a certain height, either by graphing and intersecting, tabling or using the quadratic formula. Know how to sketch the graph using at least five points spread over the paraobola and labeling your axis.
P.S. once the equation is in your calculator, table setting are 2nd window. you can change what you table skips by in there. Table is 2nd graph. Intersection, zeros and value tricks are found in the calc menu under 2nd trace and finally zoom is great for getting back to a standard window(0), fitting the graphs (fit) or making a box to magnigfy.

Chapter 4

21. Interest on a loan with Apr using Apr chart. or finding the payment on a house
22. same
23. expontential growth or decay, finding the rate.
24. predicting future values
25. interest made on compounded investment. I = M - P

Chapter 5

26. solving a system of linear equations by any means you'd like
26. something with break-even point or equilibrium point
27. some other system of linear equations with like 19-36 in 5-2

Bonus.1. maybe something like example 21 on page 225 or like 13-16 on page 225

Bonus 2. something with writing system of equations for something with an area (need to solve by quadratic equation or with your graphing calculator to find the intersection) like 226/17

That's it.

This isn't a perfect outline but Its pretty much complete. Best of luck studying with all your finals.

Mr. Rosever

Applied Astronomy Lab

Big Points.

Directions.

Neatly Number and Label all your work. Good Drawings complete sentences. You may work together but each of you is responsible for your own understanding, explanations and the original report you hand in. List your partners after your name. Do not quote them but indicate their contributions. If you need help with the charts or the pictures come see me.


On December 4th, 2005 an astronomer observed that two of the three planets circling Snoopy in the Dogstar galaxy aligned behind their sun in his line of sight. Determined to capture this image but unprepared at the time, he began observing the planets’ movements through the course of month until he was able to determine the speed of each planet. Armed with this information he calculated that the first planet, Brutus, makes one complete orbit about its sun every two years. The second planet, Hooch, takes three years to complete its orbit, while the last planet, Sadie, takes ten years, to complete its cycle. Since all the planets’ orbit were relatively circular he decided it would be best to convert this data in to a rotational speed (that is a speed given in degrees about the sun per year) to make finding the next time of alignment more easily.

1. If we break each planets orbit into 360 degrees, find the rotational speed of each planet in the units degrees of travel per year (about Snoopy) and also degrees of travel per month (you will need one or both to answer the following questions).

On December 4th, 2005, he observed that it was only the innermost and outer most planets that were aligned while Hooch was 1/6 of her orbit out in front of them.

2. Based on this information, what was the angular distance between Hooch and the other two planets on that day. Draw a picture to represent this, presenting and labeling the the planets in their proper order circling the star and entitle it "The Dogstar galaxy" December 3rd, 2005

If we use the aligned angular position of Brutus and Sadie on the day of their alignment as the reference point (the place from which we will do all measuring) it should be clear that since they are both at the reference their positions are both zero.

3. What would Hooch’s angular position be?

4. Since you have the speed of each planet you should be able to develop a chart indicating each planets’ angular distance from the reference position each month over the next year. Keep in mind that Hooch begins out in front and do so now.

5. Having done that develop another chart indicating the position each planet over the next five years skipping by a half year intervals.

6a and 6b. Use each chart to develop a linear equation for each planet that gives the planet’s location (L) as function of the time that has passed since Dec 4th, 2005. For time in your monthly equation use m (for months. For time in your yearly equation use y (for years).

7. Now you should have two equations representing each planets’ movement. Evaluate each equation to find the location of each planet, once for m= 18 and again for y =1.5. Compare your results and explain what you observe. (if you use m = 12y something cool happens too!)

8. Use your equation to find how many degrees of separation will keep Sadie from Hooch on my 30th birthday. February 4th 2007Inorder for two planets to align they must be at the same angular location L at exactly the same time. As such a system of equations can be used to solve for when and where this special occurrence of alignment will happen.

9. Use your monthly equations to create a system that can be solved to find when Hooch and Brutus will next align.Going back to the charts, we should keeping in mind that 360 degree is an equivalent position to 720 or 1080 or even 0, and 45 is the same as 405. In so doing we find that we can reduce any values for location greater than 360 by subtracting that number repeatedly until the results are within the range of 0 to 360 (0 and 360 being the same). When you make this adjustment in all your charts you will find a number of places where two of the planets are in alignment.

10. Make this adjustment then find and circle the places where at least two of the planets are in alignment.Your chart should now indicate that 2.5 years from now Brutus will again aligned with Sadie. While this is of no help to the astronomer (as the planets are perpendicular to his line of sight) it can show us something valuable when solving systems of equations like this.

11. If you test y=2.5 in both Brutus’s and Sadie’s equations you get different results that are actually the same. What are the results and Why are they actually the same?

What question 11 reveals to us is that when dealing with a cyclic system like this we can add or subtract 360 to the constant row for free to get a new, but no less valid equation and which will give a new, but no less valid result. Try these last three problems with this idea in mind.

12. Take your original equations for Brutus and Sadies location and set them up as a system to solve. Solve them. What is the result and why would it be of no help to the astronomer?

13. Take your original equations for Brutus and Sadie again and change one of the zeros in one of the equations to any multiple of 360 and try again (that is to say: since 0 = 360 in terms of angular location, add 360 to one of your equations at no cost). What are your new results and what do they mean.

14. Repeat question 13 with a different multiple of 360 (if you should get a negative be reminded that means months or years ago rather than months or years in the future but it is still okay). List your new results and explain what they mean to the astronomer?Hopefully at this point you are prepared to answer the seemingly trivial question which follows.

15. How long must the astronomer wait until the two planets he originally observed will be aligned in his line of sight (that is at either 180 or 0 with the reference) so that he can take his picture with both planets and Snoopy too.

Bonus:Having done all this work above what follows is for a five point bonus on your final exam:

Find, Explain, Show the work or the tables you used and prove to me by testing in your original equations your answer is correct for the following question.

How long must the astronomer wait to get a picture of all three doggie planets and their sun in alignment in his line of sight?


Hope you’ve had fun learning. Have a nice break. Mr. Rosever