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Pre Calculus 11 Unit 3 Summary Post

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3.3 Solving Quadratics by Factoring

First, if we want to begin solving quadratics by factoring, we must know how to factor as it is the basics of solving quadratic equations. For example, if you want to solve (2x² – 15x + 25 = 0), we must begin with factoring this equation, and that is by identifying the product of A and C and finding what two numbers make that product of A and C that also is the sum of B. Such as, in this case, A = 2, B = -15, and C =25, there are two methods in factoring this which is either the X method or the square box method, personally I prefer the X method. The factors of 50 (because A • C = 2 • 25 = 50) is 1 • 50, 2 • 25, and 5 • 10. Which one of these has a difference or sum of -15? 5 and 10, in this case, it would be -5 and -10 because (-5 • -10 = our (ac) value of 50 as well as it is the sum of our (b) value of -15 (-5 – -10 = -15).) Next, once we have these values we can use the X method to factor,

after we have placed our values, we can divide the left and right side by the A value

Yes the 1 underneath the 5 is unnecessary but personally, it helps me visualize the factor more easily. Then, to factor it is (2x-5)(x-5), personally, how I usually distinguish which numbers go where is if you look at the numerator of each fraction which is (-5 and -5) you can recognize the plus or minus symbol which is a minus in this case and if you compare it to the recipe of a factored quadratic which looks like (x -/+ #)(x -/+ #), you can sort of visualize the placement of the numerator value into a factored form from matching the symbol (+ -) placement. Another way to remember is bottoms up which is to find the denominator (bottom) and moving it up which makes it first and before the numerator when writing them into two factors in brackets. After that, we can find which values would make it so the equation would equal 0. For (2x-5)(x-5) X = 5/2 and X = 5 because (2 (5/2) – 5 = 0) and (5-5 = 0)

Once we know how to factor we can finally begin solving quadratics. A question that looks different but uses the same formula of factoring quadratics is (2x – 3)(x + 1) = 3. First, the right side must equal 0 and the left side must be in general form (Ax²+Bx+C), so expand the left side and simplify,

2x² + 2x – 3x -3 -3 = 0

2x² – x – 6 = 0

Factor trinomial

(2x + 3)(x – 2) = 0

Then, to solve, 2x + 3 = 0 -> 2x = -3 -> x = -3/2

AND x – 2 = 0 -> x = 2

After we can verify the solution,

(2(2) -3)(2+ 1) = 3

(1)(3) = 3

3 = 3

AND

(2(-3/2) – 3) (-3/2 +1) = 3

(-6)(-1/2) = 3

3 = 3

Next, we can try an equation that involves a common factor, one example could be (2x² + 18 = 12x)

First we must move everything to the left side so the equation could equal 0

(2x² -12x +18 = 0)

Then remove GCF which is 2 because the value of A, B, C in the equation can be divided by 2 (2,-12,18)

2(x² -6x + 9) = 0

Then factor

2(x – 3) (x – 3) = 0

X = 3, X = 3

Verify

2(3)² + 18 = 12(3)

18 + 18 = 36

36 = 36

One more simple example that is not quite the general form but still uses the same properties of solving quadratic equations is

(2x²=4x)

Move 4x to the left side to make the equation equal to 0

(2x² – 4x = 0)

Factor the polynomial

(2x (x – 2) = 0)

X = 2 and X = 0 (x = 0 because if you put 0 in the x in 2x then it would be 2(0) and 2 •  0 = 0)

Verify

2(2)² = 4(2)

8 = 8

AND

2(0)² = 4(0)

0 = 0

You can also apply this strategy of solving quadratic with radical equations, such as (√23-x = x – 3), we must rewrite this equation into Ax²+Bx+C = 0 form, to do that we first isolate the radical, which makes the equation to,

(√23-x)² = (x – 3)²

we square both sides to remove radical

(23 – x = (x – 3)(x – 3) )

Then expand the right side

(23 – x = x² -6x + 9)

Move the left side to the right side to make the equation equal 0

(x² – 5x – 14 = 0)

Factor

(x – 7) (x + 2)

X = 7, and x = -2

Verify

(√23- (7) = (7) – 3)

(√16 = 4)

4 = 4

AND

(√23- (-2) = (-2) – 3)

(√25 = -5)

5 ≠ -5

As you can see all the questions before this the solution worked when verifying, however, when it comes to radicals the solution may not always work, which is why it is very important to verify especially when dealing with radicals. This means that (-2) in the example above is an extraneous root, where it may seem like the solution would work because you factored it, but it actually doesn’t and is false.

An example that uses the method of solving quadratic but also incorporates a method of the previous chapter of substituting variables is (2x-1)² – 2(2x-1) – 8 = 0, this looks like Ax² + Bx + C = 0, we can replace (2x-1) to W, which makes the equation w² – 2w – 8 = 0 then we factor, (w-4)(w+2), then we replace (w) with (2x-1), which equals to (2x-1-4)(2x-1+2), simplify, (2x-5)(2x+1), then x = 5/2 and x = -1/2, verify, also when verifying, make sure you input the solutions in the original equation which is (2x-1)² – 2(2x-1) – 8 = 0, to make sure the answers are accurate

X=5/2:(4)² – 2(4) – 8 = 0, 0 = 0, AND X=-1/2: (-2)² – 2(-2) – 8 = 0, 0 = 0.

Lastly, we can use a quadratic equation to solve a problem, this makes you think differently as you need to know which numbers are needed and where they are needed to be put, instead of just doing equations already given to you.

Such as, the area of a rectangular sheep pen is 96m². The pen is divided into two smaller pens by inserting a fence parallel to the width of the pen. A total of 48 m of fencing is used. Determine the dimensions of the pen.

The area of the pen is 96m² and the width is W in metres, to find the length it would be 96/W = L, which are the dimensions. The total length of the fence is 48m so an equation would be (3w + 2(96/w) = 48) First we multiply each term by w to remove the w underneath the 96, (3w² + 2(96) = 48w), make it general form, (3w² -48w + 192), next we divide each term by 3, 3(w² – 16 + 64), then factor, 3(w – 8)(w – 8), so (w = 8), lastly 96m²(area)/ 8m(width) = 12m(length), so the sheep pen has a length of 12m, a width of 8m, and a area of 96m²

In conclusion, solving quadratic by factoring is simple and some important things to remember and to do is when solving equations are, know how to factor quadratics, know how to make it into general form (Ax² + Bx + C= 0), verify the solution (especially radicals), and your main goal is to know X or what the variable equals to; to make the equation true.

Pre Calculus 11 Unit 2 Summary Post

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Adding and Subtracting Radical Expressions:

To start we can remind ourselves how to simplify and work with like terms, the strategies for simplifying polynomials can be used to simplify sums and differences of radicals. For simplifying polynomials an example could be (3x + 5x =), since the variables are the same (x) then we can add the coefficients which equals to (8x). For radicals, what must be the same in order to simplify each other is the radicand and index, such as, (4√2 + 3√2) can be simplified to (7√2). Another example could be (5√2 + 3√18 – 4√8) Although it may seem like the radicands are different, you can simplify the radical (3√18) into (9√2) and (4√8) into (8√2) then we can simplify that into (5√2 + 9√2 – 8√2 = 6√2). Also, it is important to notice that the indexes are the same, which is 2 in this case, if not, you cannot simplify the radicals such as (3√3 + 4∛3) because even though they have the same radicand the indexes are different at 2 and 3. That being said, once we have like terms (radicand and index) we can simplify them by combining the co-efficient, and ultimately rewriting the equation as the most simplified result as it can be. To add, knowing the prime factors of numbers and how to simplify radicals are really important as it is basically the foundation of doing these types of questions like if you don’t know how to find the prime factors of a number you cannot fully simplify the radical making it really difficult to complete a question.

We can try a question that is a little harder but still builds with the knowledge and rules we know.

Simplify (√20x⁵y⁴ – √125xy²)  First we know the indexes are the same so that’s good but the radicands are not, so the next step is to simplify 20 and 125 which makes the expression into (2√5x⁵y⁴ – 5√5xy²), next are the variables, for (2√5x⁵y⁴) 2 groups of 2 of x⁵ can move to the co-efficient because the index is 2 which leaves 1 x remaining in the radicand, for y⁴ 2 groups of 2 can move with the co-efficient which is all the y’s because 2 x 2 = 4 and it is y⁴ (y⋅y⋅y⋅y). We can apply the same with (5√5xy²), in this case, only 1 group of 2 of y can move next to the co-efficient because it is y², the reason for being groups of 2 is due to the index (√ no number means 2) if it was ∛ then it would be groups of 3. After simplifying the radicand’s it brings it to (2x²y²√5x – 5y√5x) As you can see the radicands are the same so we can simplify to ( (2x²y² – 5y)√5x) ) TIP: If you are not sure if you did it right, just expand the simplified radicals and see if you got the original expression.

One harder question is (5√8x³ + 4y√75y³ – 2√27y⁵ – 3x√50x + 4∛3y)

First, simplify the numbers in the radicand

Then the variables

Combine like terms

Simplified!

Along with all this, we can also identify the values of the variable after simplifying and there are some rules, one rule is how the radicand cannot be negative if the index is even, for example, 5√x + 3√x – 4√x = 4√x, since the index is 2 the radicand cannot be negative, meaning each radical is defined for x ⩾ 0, or else it would be called “undefined” or “non-real number”. However, if the index is odd like 5∛x then it can be defined as x ∈ R (any real number). If it is something like q∜p³q, then to be defined, p⩾0 q⩾0 or p⩽0 or q⩽0, because if both p and q are negative in the radicand then the outcome would be positive which is why a negative could be possible in the radicand even with an even index as long as it comes out as a positive. Another scenario when there could be negatives in the radicand with an even index is if a question were to be 2√2a², this would be (a ∈ R) because of the exponent of 2, for example, if a = -2, it would become 2√2 (-2)(-2), which equals to positive 4 (2√2(4)) -> 2√8 -> 4√2 which would work and can be defined as (a ∈ R) because both a negative or positive can replace (a). But it would not work if the exponent in the radicand was odd because the end result of the radicand would be odd and that is not allowed. Overall, the end result of the radicand before must be positive (greater or equal to 0) if the index is even for it to be defined, and if the index is odd then the variable is equal to all real numbers.

Fight Club Museum

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Final Project For Lit Circles By:

Jonathan, Hudson, Shanez, Cleon, Wyatt, Daniel

The Physical Exhibit Space

When you first walk into the museum, you are surrounded by explosions, yellow/orange colored walls with a gun on display. This room represents the star of our novel when we are introduced to Tyler and the nameless narrator who are both on top of a burning building with Tyler holding a gun in the narrator’s mouth. The second room right in front represents the narrator’s kitchen with Swedish furniture that he is very fond of, the checkered wall represents the “game of checkers” going on with his split personality. The green room on your left with chairs is the support group that the narrator goes to for his personal healing with the quote “Faker. Faker. Faker” to represent Marla also faking her sickness to attend. The next half of the room with white bricks represents Fight Club itself, the rules that go along with Fight Club and the plot diagram, the second part has Tyler’s car where the narrator and Tyler stay when Marla is at Tyler’s house. The next room to the left of the kitchen is the room with visuals of the main characters. The next room is the gift shop where you can buy specialized Fight Club merch, soaps and  copies of the book. The second last room represents Tyler’s house on Paper Street where he starts Project Mayhem and where he has “space monkeys” making soap and preparing for the project. The last room of the museum represents the end of the story when the narrator (who we discover is Tyler Durden) shoots himself (the gun) then ends up in “Heaven” but in reality he’s in a hospital and the meds add on to that fact, there’s also a picture of half of the narrator’s face merged with Tyler’s to emphasize that Tyler and the narrator were the same person from the start.

The Museum: 

The Fight Club Merchandise

 

Merchandise #1 (Front and back of shirt)

This shirt represents the story because it has a fight club soap on the front which is the name of the book and has soap which was a big part of the story because it was their main source of income. The back of the shirt represents the story because it has the first two rules of fight club on it. These are probably the most important rules because it shows how secret the fight club was and now how big it has become.

 

Merchandise #2 (Front and Back)

This t-shirt has the chemical formula to create dynamite in the story. This is important because dynamite is used widely throughout the plot in different scenarios. Such as when the narrator’s condo was blown up using this particular dynamite, or when Tyler says he can blow up bridges and buildings using this homemade dynamite. Tyler even mentions that with enough of these ingredients, you can basically blow up the entire world. This same chemical formula for dynamite was used at the end of the book when the Narrator and Tyler were on top of the building and Tyler threatened to explode the building. However, the bomb did not detonate and malfunctioned because Tyler mixed paraffin in it and this final plan from Project Mayhem had failed. At the end, the space monkeys from Project Mayhem tells him the plan continues and expects Tyler to come back, meaning that the narrator has not gotten rid of Tyler.

Merchandise #3 (Soap)

This bar of soap represents the novel Fight Club because in the novel, the main characters owned a soap company. The soap company was named Paper Street Soap Company which is integrated in this bar of soap. This bar of soap gives the buyer the feeling of owning the homemade soap that the narrator and Tyler Durden created. They created the Paper Street Soap Company to be the main source of income which allows them to invest more in fight club and project mayhem which are the two main groups in the novel. Tyler mentions in the novel that to make the soap, he uses Marla Singer’s mother’s collagen as the fat source for the bars of soap. When Marla’s mother would have a liposuction Marla would save the extra fat in case her mother needed more collagen injections. Tyler would store this fat in several bags in the freezer and the Paper Street Soap Company makes a fortune off this soap.

The Artifacts

1.  The Bed

The bed is a very significant item as it is an important symbol in the novel. The bed is a symbol for the narrator’s insomnia as he does not get any sleep. When the narrator falls asleep, Tyler Durden is off doing something crazy that someone has to be out of their mind to do. The bed also symbolizes the fact that the narrator suffers from insomnia. It is important to know that the narrator has insomnia as it is something that affects him drastically throughout the story. The narrator’s insomnia essentially causes the narrator to never feel as if he is awake or if he is asleep. This insomnia develops into the narrator discovering his split personality of Tyler Durden which completely changes the narrator’s mind and thoughts. This all leads into the narrator attempting to find Tyler which is his split personality and he plans to take the role of the leader and shut down the fight clubs.

2. The Explosives

The explosives are significant as they are used a lot in the story. Explosives are used mostly by Project Mayhem to take down buildings and destroy modern civilization. Another reason explosives are important to the story is because it is a bonding item over Tyler and the narrator Tyler teaches him how to make explosives and it sticks to the narrator as he is always mentioning explosives and him knowing it because of Tyler. Explosives are kind of a symbol for Tyler as he is extreme, bold, and mischievous just like explosives would be.

3. The Space Monkeys

The space monkey in the book represents how society is easily manipulated into sheep. In the book, the space monkeys are workers in Tyler’s Project Mayhem. They are recruited from Fight Club and they are trained to do simple tasks over and over again. The tasks they do are pushing buttons, boiling fat, cooking food, gardening, etc. They must shave their heads and wear all black which makes them look all alike. This symbolizes how people in society blindly follow leaders even though they do not know what they are working towards. They suffer mentally and physically all for a paycheck just like how modern society is in the real world. They are sheep controlled by people higher to make them feel important but in reality are just doing simple tasks. And their fingerprints get burned off with lye to make them even less special. They also must be obedient and blindly follow their leader and if they disobey, violent consequences will occur.

4. 68 Chevy Impala

The 68’ Chevy Impala is significant to the story because it is where the narrator and Tyler would go when they want to just talk and have a couple of beers. The Impala is also a place where the narrator and Tyler go when they need to lay low from Marla or to lay low from their home. The Impala was chosen by the narrator and Tyler because it has very large front seats so if they ever needed to sleep in on a Saturday night, they are able to. Another reason why the Impala is significant to the story is because it is where Tyler tells the narrator about how to make soap along with what fats to use and what fats not to use. Overall, the 1968 Chevrolet Impala is significant to the story because it was the place that the narrator and Tyler went to hang out and it was the place where Tyler told the narrator about how to make soap properly.

5. Testicular Cancer Poster

The testicular cancer poster is significant to the story because it represents where the narrator meets some important characters. The narrator meets bob at the testicular cancer support group after he had gone to many different support groups with many different names. The narrator also meets Marla at the different support groups because she was doing the same as the narrator and going to many different support groups. This is also where Marla calls the narrator a “faker” and accuses him of not having anything wrong and lying to all the members. Another reason why the poster is significant is because the support groups are where the narrator felt he belonged because he felt his life could not get any worse just like the support group members. Overall the testicular cancer poster is significant because it represents where the narrator met characters like Marla and Bob, also because it is where the narrator learned that his life could get worse even though he felt like he fit in.

Character Profiles

                                                                    

Tyler Durden Tyler Durden met the Narrator during a business trip on a nude beach. After the Narrator’s apartment exploded Tyler invited the narrator to stay with him in his house and Soap factory. Tyler is the polar opposite of the narrator, where the Narrator works a steady job, Tyler makes soap when he wants to, where the Narrator lives in a nice apartment, Tyler lives in a run-down house, where the Narrator dislikes Marla, Tyler is in a relationship with her. Tyler and the Narrator’s friendship lead to the creation of the Fight club, and eventually Tyler branches the fight club off into his “Project Mayhem”.

                                                     

Marla Singer This is Marla Singer, Marla Singer is an acquaintance of the Narrator and in a relationship with Tyler. Marla met the Narrator when they both went to a disease support group and the narrator realized that she must be faking it. This led to the different support groups throughout the city being split up among them. Marla met Tyler when she called the soap factory wanting to talk to the Narrator about how she was planning on killing herself, this led to Tyler picking up the phone and going to see her. This led to Tyler starting an intimate relationship with her. Nearing the end of the story Marla reveals information to the Narrator that changes his outlook on reality.

The Setting Of Fight Club

There isn’t a specific setting for this novel as no city is mentioned in the book. We can predict that the story takes place in a city in the United States of America because of the other nearby cities mentioned in the book. The book first starts off with the two main characters on top of a building that is blowing up but, later we get a glimpse into the narrator’s life (his house, the support group, his workplace.) We also see that there isn’t much use of technology in the book besides computers which gives us an idea about the year that the story takes place, (the first laptop/portable computer was made in 1975) around the 70’s and 80’s. The two main characters also spend time together in bars and underground bars where Fight Club is invented which were also main hangout spots in those times. Also we know that the story is fairly closer to our time because of the expensive cost of soap ($20) which is still relevant nowadays. Tyler’s apartment on Paper Street is also the setting for project Mayhem and is also a relevant place where main events happen. They also make soap by boiling fat the old fashion way and by making individual pieces of soap, this is represented in the Paper Street Soap Company room in our museum where there are no machines or factories to make soap.

Our Ad Campaign

The admission cost of our museum will be set at $20 with an optional fee of $280, the admission is set at this price because $20 was the cost of one of Tyler’s bars of soap and $300 was the cost of becoming a space monkey. Basic admission the the museum comes with a free bar of soap, and advanced admission comes with the opportunity to join project mayhem and become a space monkey. This museum is for people who are mature and not affected by any disturbing content. These advertisements will be played across various different movie channels to entice a mature audience for Fight Club.

What I Accomplished In This Project

In this final project, I completed areas of the projects regarding the artifacts, merchandise, the commercial, and other minor parts. For the artifacts, I completed the plot diagram about the novel that was needed for one of the artifacts and the Space Monkey artifact paragraph. For the merchandise, I designed the Fight Club t-shirt, and co-designed with Hudson the Fight Club rules sweater and dynamite t-shirt. As well as writing the merchandise paragraph about the chemical formula for dynamite t-shirt. I also found two pictures for the character profile part, which I thought one of them really helped depict the circumstances and idea about where they stand between the three main characters in the novel (like how Marla is physically and mentally between Tyler and the narrator). For the actual museum, I think everybody contributed ideas for it but mainly Shanez and Daniel worked on it. For the commercial, I didn’t play a major role but the part I played was the introduction/explanation of the plot of the novel, as well as some behind the scenes work and ideas, such as the soap fade and soap chuck at Hudson.

Fight Club commercial:

 

 

Engineering 11 Bridge Reflection

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Bridge Reflection

In this engineering project, we had to create a wooden bridge that will span across a 30-inch gap. It also had to be at least 5 inches wide and individual pieces had to be 1/8 of an inch thick, and of course the bridge must also be at least 30 inches long.

What we did well was the idea of gluing multiple 1/8th pieces of wood together to reinforce each individual pieces of our design. This way each part of our design is stronger and sturdier, instead of the flimsy 1/8th piece. I also thought our design was pretty good, it had 2 support wood pieces in the middle, underneath the main board and on top of it. This supported the middle point which prevent it from bending and snapping. We also added a piece of wood going straight up with strings attached from the bottom which helped the bridge by exerting the tension of the weight. That really helped our bridge hold at least 334lbs.

The areas we struggled and could have done better was the gluing of the individual 1/8th pieces. In some parts, the glue was uneven making the bridge have gaps in between the 1/8th pieces. What we could have done better was applying an even layer of glue and evenly apply pressure using the clamps. So that the pieces would have a flush finish with no gaps.

What I would change if I could go back was letting the glue settle before we continued. When we glued the middle pole piece on our bridge, the glue had not settled before we attached the strings. Meaning that the tension of the string on the very first one we attached, was already pulling one side. If the glue had settled, then the middle pole piece would be sturdy in that position and would not move when attaching the first tensioned string piece. Another thing I would go back and change was when we tested it. Although I wasn’t there, I would have continued to added weights on our bridge until it collapsed. So, we could see the maximum weight our bridge can hold, but unfortunately we stopped at 340lbs.

Engineering 11 The Ultimate Cardboard Box Reflection

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The Ultimate Cardboard Box

In this project, we had to create a cardboard box that is made for shipping. First, we watched a video about another box made called the Rapid Packing Container. Then we compared it to the traditional box used commonly in shipping companies such as Amazon. We then had to analyze the pros and cons within both designs to create a prototype. What was ideal for me was that it had to be easy to open, waste efficient such as minimal tape, and strength/durability. For me, I liked the jigs they used in the Rapid Packing container, as well as the flap method in the tradition box. After brainstorming a few ideas and drawing a couple designs, I came up with this.

After creating my first 4×4” prototype, I realize there could be a few adjustments made on my box and design. First, I noticed that my cuts were too deep making the cardboard weaker on the folding points. Second, I had to resize my flaps supporting the walls as it was too small making my box compress. Lastly, I needed to cut the slit for the jigs bigger because it was too small. I slowly made bigger prototypes like 5×5 inches and 8×8 inches. Along the journey, I realized folding outward was stronger than folder inward as the paper side of the cardboard supported the edges more. After perfecting my prototype, I had to make my final product, but I ran into another obstacle. The pieces of cardboard available to me was too small for my 12×12 box. So, I had to improvise by using two pieces of thicker cardboard that was long enough to have two separate pieces of my box. After cutting out two separate segments of my box design, I used smaller piece of cardboard to attach them together, acting like a spine of a book. (also included new flaps)

The pro’s of my box are that is uses less tape compared to the traditional box, the strength and durability, easy to open (doesn’t require a knife or tool), and easy to pack and reuse. A con would be that it does not lay fully flat when unfolded with some flaps flaring up, and also it is heavier and denser as a trade from durability and strength.

I used many different tools and materials to construct this box, such as, ruler, pencil, X-actoknife, cardboard, tape, hot glue, and eraser.

What I learned in this project is that it takes time and effort to create something. There is a process you need to endure when you take on a project. You cannot expect things to go well on your first try, and that you will have obstacles and problems along the way. Most importantly, I learned how to efficiently cut cardboard and how to score the cardboard to appropriately fold to your liking.

What I’ll improve next time is when cutting up the cardboard, I will cut more evenly and straight so that the edges of the box 100% line up with each other.

I would give myself a B+ because throughout the weeks working this project, I had been effectively using my time to improve and perfect my box. Which I thought was good because that is very important when creating a new project from your own design. What I thought could done better is when drawing my design on the cardboard, I could’ve drawn it more precisely and correctly, that way I would have had less tries. On the other hand, if I haven’t had make so many boxes, I wouldn’t of know the flaws and adjustments that led to perfecting my box.