Week 11 – Graphing Inequalities and Systems of Equations

Posted on by shannonf2015

For the past week, I have learned how to write solutions for quadratic inequalities and also how to graph them.

Let’s start with an example:

Example 1: Given an equation

2x^2-8x-10<0

First step is take out what is common then factor to be able to find the x-intercepts.

  1. Common: 2

2(x^2-4x-5)<0

2. Factor

2(x+1)(x-5) < 0

x-intercepts: -1 and 5

Now that you have the x-intercepts, we can draw these on a number line.

By looking at the equation, we can tell the parabola will be opening up as the coefficient next to x^2 is positive. So there is a +, then a – , then a + as the parabola will dip past the x-axis since it has two x-intercepts.

Looking at the inequality symbol, it’s using <. This means they are asking for the numbers that are less than 0, so negative numbers.

 

 

The negative numbers are in the middle so the solution would look like this:

-1 < x < 5

The circles are not coloured because it is not including “or equal to”. This equation is just using less than.

Now let’s look at another example where we are given a graph and we must write an inequality.

Let’s go over some important things:

≤ and ≥ : has a solid line

< and >: has a dash line

Example:

First step is to find the y intercept, which is -1. Next is to find the slope. We can tell by the points that it’s going up by 4 and moves to the right once. So the slope is 4. This linear graph is tilted up so it’s positive.

The equation so far looks like this:

y ☐ 4x^2-1

To find the inequality symbol, we must test a number from the shaded section that is not a point on the line and determine which symbol to choose which will result in a correct answer.

Let’s use (5,0)

Plug in 5 in x and 0 in y:

0 ☐ 4(5)^2-1

0 ☐ 4(25)-1

0 ☐ 100-1

0 ☐ 99

To make this true, we must use ≤ because 99 is greater than 0 and the line is solid. 

So the equation will be:

y ≤ 4x^2-1

 

 

Week 10 – Radicals, Arithmetic Series and Infinite Geometric Series

Posted on by shannonf2015

During review, I have recalled some concepts I have forgotten.

One of them is when a value or variable has an exponent that is a fraction. When it’s a fraction, the denominator represents the root. We can remember this as Flower Power.

Example:

64^\frac{1}{3} —> \sqrt[3]{64} —> 4

Another simple method I recall is adding an arithmetic series that is finite. The first step is taking the first term and the last term in the series and adding them together. Next, count how many terms there are in total and divide the amount by two. Multiply the terms divided by two with the sum of both first and last terms.

Example:

10 + 5 + 0 -5 -10 -15

  1. 10 + (-15) = -5
  2. Number of terms in total: 6
  3. \frac{6}{2} = 3
  4. (-5)(3)
  5. Sum = -15

Another concept I did not talk about last time was infinite geometric series. We can tell if they converge or diverge by looking at the value of the common ratio.

Diverges: r > 1; no sum

Converges:  -1 < r < 1

Formula:

Image result for infinite geometric series formula

Example:

2 + 3 + 4.5 + 6.75 +…

r = \frac{3}{2}

The common ratio is greater than 1, so it is infinite (diverges); no sum.

-0.5 – 0.05 – 0.005 – 0.0005 – …

r = \frac{-0.05}{-0.5} = 0.1

The common ratio is greater than -1 and smaller than 1 so it is finite (converges); has a sum.

Let’s look at an example of how to use the formula with a geometric series.

8 + 2 + 0.5 + 0.125 + …

Information given:

  • a = 8
  • r = \frac{1}{4}

Plug in the information given in the formula and solve.

Remember to flip the fraction (reciprocals) when doing division.

 

 

Week 9 – Quadratic Functions

Posted on by shannonf2015

Throughout the week, I have learned how to convert a quadratic function in general form to standard form and factored form to be able to find different information about how the graph looks like.

Let’s start with an example:

y=2x^2-20x+32

Right away, we can tell that the y-intercept is 32 because it is in general form.

We can also tell the direction of opening as it is positive, which means it opens up and has a minimum value.

Start by removing the greatest common factor which is 2.

y=2(x^2-10x+16)

Next, factor the three term in the bracket to find the x-intercepts (the roots)

y=2(x-2)(x-8)

x intercepts: 2, 8

Now we need to find the vertex, to do that we need to change the general form to standard form by completing the squares. We cannot change from factored form to standard form.

y=2x^2-20x+32

First take out what is common between the first two terms, leaving the 32. Leave two spaces for a + value and – value between middle term and last term to create zero pairs.

  • y = 2(x^2-10x+___-___) + 32

Take middle term and divide it by 2 and square it.

  • -\frac{10}{2}=5 —> (-5)^2=25
  • y=2(x^2-10x+ 25-25) +32

Multiply the coefficient (2) with the negative number of the zero pair (-25). Then do the calculations with the last term.

  • y=2(x^2-10x+25)+2(-25)+32
  • y=2(x^2-10x+25)-50+32
  • y=2(x^2-10x+25)-18
  • y=2(x-5)^2-18

Standard form: y=a(x-p)^2+q

Vertex: (p,q)

The vertex is (5, -18); the value of p changes sign when written for the vertex.

Now that we know the vertex, we can determine the axis of symmetry, domain and range.

AOS = x=5

Domain: xER

Range: y ≥ -18

This equation is also congruent to y=2x^2.

Another example is when we are given some information about the quadratic function and we must write an equation.

Example: The graph of a quadratic function passes through (1, -7) and the zeros of function are -6 and -1. Write an equation in general form.

Information given:

Point: (1, -7)

X-intercepts:

  • X1 = -6
  • X2 = -1

By looking at the information, we cannot use the standard form because we are not given the vertex.

We therefore have to use factored form as we have x1 and x2 as well as a point.

Factored form: y=a(x-x1)(x-x2)

Replace

  • y = -7
  • x = 1
  • x1: -6
  • x2 = -1
-7=a(1+6)(1+1)

When x1 is replaced with -6 and when x2 is replaced with -1, the sign changes to positive because there are two negatives.

Next is to isolate a.

  • -7=a(7)(2)
  • -7=14a
  • \frac{-7}{14}=\frac{14}{14}
  • -\frac{1}{2}=a

Now that you have a, put it in the equation. Remember, they want it in general form.

  • y=-\frac{1}{2}(x+6)(x+1)
  • y=-\frac{1}{2}(x^2+7x+6)
  • y=-\frac{1}{2}x^2-\frac{7}{2}-3

 

 

 

 

Narrative Essay: The Torture of Living with my Childhood Friends

Posted on by shannonf2015

Honestly, there is not much I am proud of in this essay but if I had to pick, it would be my word choice and how I implemented the dialogue. Instead of telling the story, I was able to use imagery to show the story and to describe my feelings throughout this experience. I am also proud of how I was able to put dialogue throughout my essay in a way that flows through the story.

I need to improve on how I end and resolve my conflicts. I need to improve on not ending the story too abruptly and to add more details about what has happened after the climax and falling action. Another thing I need to improve on is to make my stories more engaging and to have better transitions.

 

The Torture of Living with my Childhood Friends

“No, I don’t want to be with her! Why don’t you?” exclaimed Samantha.

“No, I’m always partnered with her!” argued Tiffany.

These were the people who I thought were my friends. The ones who knocked on my door to go ride our bikes, the ones who ate lunch with me, and the ones who invited me to their birthday parties. We stuck together like strong hydrogen bonds, constantly following each other like a girl’s club. I was the one who tagged along no matter how nasty their words burned.

There was four of us in the group: me, Samantha, Makenna and Tiffany. We hung out everyday at school; our lockers were perfectly aligned, and our flawless synchronized walking made it more believable we were the ultimate dream squad. Every movement and opinion became a trend; if one of us didn’t follow, they would be crucified by the group’s judgemental eyes. Music choice, singing voice, and dancing skills were all the requirements you needed to be accepted in the squad. Big Brain? Forget that. All you needed were some ears, a pair of eyes and a neck to do all the nodding to become one of their minions.

To make it even more cheesy, we wore heart necklaces which represented that we were BFFLs (Best Friends For Life)! Beyond cooler than the standard BFF. Samantha, the one I knew since kindergarten, convinced me to change my misfortunate diy hairstyle, cut by my own mom, to growing out my bangs so that I can become “pretty” at the age of seven. I began to grow my bangs and wear only the clothing we bought together during our shopping trips to the mall: black leggings, stylish pastel coloured t-shirts, and some converse shoes.

Whenever we were together, they did most of the chatting: fun activities they did over the weekend, inside jokes I didn’t understand, and boasting about something in their life. I was the loner of the group. I couldn’t get a single word in the conversation, with my questions and comments drowned by their voices, while feeling useless.

You know the stereotype Asian people get about having small eyes? Well I was one of those victims, by my own friends. While we were waiting for our parents by the school under a big oak tree, Samantha pointed towards another girl named Olivia, one of my classmates.

“Hey! That’s Olivia. You guys look kind of alike,” Samantha says as we lay on the patchy prickly grass.

“How?” I asked.

“You both have small eyes.”

If she couldn’t be even more offensive, she stretched her eyes out with her fingertips, making her eyelids a thin line as her eyes. Everyone laughed, and I awkwardly laughed as well.

The next day, I met up with my friends before class started. They were chatting about some guy they all liked, with their flirtatious googly eyes while twirling their perfect shiny hair; I could almost picture arrow hearts piercing through their chest. If only that really happened.

“I think he likes us,” stated Samantha.

“Definitely,” added Tiffany in agreement.

Samantha’s eyes turn towards me with her nose scrunched like I was a piece of old gum under her shoe.

“But I don’t think he likes Shannon,” added Samantha.

My self esteem instantly went down the drain along with my shattered heart. From that day on, I became self conscious, only focusing on my impurities. Not only did they rank me the “least prettiest” in the group, they criticized my eating habits, my clothing and my personality.

During summer in grade eight, we planned to go to a show to see our favourite youtuber, Tyler Oakley. I was stoked to watch my artificial dose of sunshine live without my phone screen. When the day arrived, I jumped out of bed with a sleepy widened smile, while grabbing my new neatly laid out clothes. I slipped on the black ruffled skirt covered in daisies and my silky ocean blue top, gliding my hands across the soft fabric and putting on my silver heart necklace given by my parents. The cold metal pressed against my skin, cooling down my excitement that was flooding through me.

As I walked towards the building where they held the show, I noticed in the corner of my eyes Samantha, Tiffany, and Makenna striding across the street with their dazzling matching outfits and their gracious hand gestures that flowed through the air like a ballerina. It was entrancing, however, I had to stick to my roots and make my move. By the time I caught up to them, with the sun making my hair greasy and sweat dripping down my forehead, the first words that came out of my mouth was a croaky “hi”.

“Hi, do you have your ticket?” asked Samantha with a fake smile.

“Yeah! Right here!” I responded with excitement.

“Cool, let’s get in line then.”

When we got in line, filled with nauseating screaming girls surrounding me, behind, I heard whispers between Samantha and Tiffany.

“What is she wearing? That skirt is so short I can see her underwear!” whispered Samantha.

“I know, who does she think she is?” added Tiffany with a sneer.

“How did her mother let her out like that?”

The words took a punch to my stomach, making me unbalanced. My face started to heat up from embarrassment; therefore, I decided to tie the quilted sweater I took in case it gets cold, around my waist. Relief swarmed through me like bees buzzing back to their hive once I was completely covered. When we entered the building, bright lights and loud music blasted through the room. The set up was like a movie theater, but with more levels of seats and every seat filled with a screaming teenage girl. It was horrifying and awesome at the same time. Our tickets held our numbers which determined our seats. We sat on the third level; perfect view from above. When we sat down on the red velvet cushioned chairs, Samantha started making eye contact with Tiffany. They spoke to each other during the thirty minutes before the show started. Since I was sitting between them, my body was bending back and forth like a ping pong ball in my seat while they spoke, bending in front of me and behind me. When will they stop, I thought. I wanted to say something, but I didn’t want to upset them. Therefore, I stayed quiet, swaying back and forth like riding on a seesaw. Five minutes before the show, the lights were getting dark, signaling Tyler’s arrival! I wiggled my bum in my seat with anticipation, when Samantha asked Tiffany a favor.

“Tiffany, can you move seats with Shannon?” asked Samantha.

“But I like my spot, no one tall is blocking my view. I’m the shortest out of all of you,” I protested.

“Just move!”

The only thing I could think of was move. What else could I have done? Yell back at them in public? I was already afraid of speaking and getting kicked out of the group, leaving with no friends. I switched with Tiffany, with my view now blocked by a middle aged, tall, bald man; probably the father of the daughter sitting beside him. Makenna noticed I was awkwardly stretching my neck like a wannabe giraffe, so she kindly told me to switch spots with her. Although I was completely shattered and devastated, it was quickly replaced with happiness when Tyler Oakley walked on stage.

When the show ended, Samantha and Tiffany left with their parents without saying goodbye. Makenna and I shared an awkward hug and our day ended there.

Near the end of summer, it was time to get ready for High School! The first day of transferring to High School was not so bad. I may have walked into the wrong class room, spilt tea on my new top and tripped into a garbage can, but I managed to go a day without seeing my friends. From that day forward, I decided I didn’t need them anymore.  I met up with new friends, who are my friends till this day, who are accepting, honest, and trustworthy. I didn’t have to act anymore, just be myself.

Despite feeling ignored and mistreated by my frenemies, a valuable lesson came out of this experience. We shouldn’t feel the need to stick with those that have hurt us physically or verbally even though we’ve kept a long friendship. People continue to come in and out of our lives and some may re-enter, but true friends will always be part of our lives.

 

Week 8 – Quadratic Functions and Graphs

Posted on by shannonf2015

I have learned many things about graphing such as what information we can take out to determine how a graph looks like of a quadratic function. I have also learned the most important information to determine the equation of a quadratic function and how to draw it as a graph.

Parabola: graph of every quadratic function which is a curve

General Form: y=ax^2+bx+c

Standard Form: y=a{(x-p)}^2+q

Vertex : highest or lowest point; (p,q)

Minimum Point: when the graph opens up (The coefficient next to x^2 is positive)

Maximum Point: when the graph opens down (The coefficient next to x^2 is negative)

Axis of Symmetry: intersects the parabola at the vertex

Domain : All possible values for x (always the element of real numbers)

Range: All possible values for y

X intercepts: zeros of function (values of x when the function is 0, when y=0)

Pattern of parent function, y=x^2: 1,3,5,7,9…

Throughout the week we have different types of transformations:

Parent Function : y=x^2

y=x^2+q : depending on the value of q, the image of the graph y=x^2 moves a vertical translation (moves however many times up or down) while keeping the same points, size and just sliding the vertex. This quadratic equation is therefore congruent to the parent function, y=x^2, because it does not change, it just moves up or down.

y={(x-p)}^2 : depending on the value of p, the image of the graph y=x^2 moves a horizontal translation (however many times to the right or left).

  • {(x-3)}^2 —– when the sign is negative, the vertex of the graph moves to the right
  • {(x+3)}^2 —– when the sign is positive, the vertex of the graph moves to the left

This is also congruent to the parent function.

y={ax}^2: the graph will stretch vertically when a > 1.

When the value of a is between 0 and 1 (a fraction), the graph will compress vertically : 0 < a < 1

This transformation is not congruent to the parent function.

When these transformations are all combined, the equation becomes standard form. 

Let’s look at some examples of using the standard form to determine an equation of a graph.

 

1) Vertex: look at where the highest or lowest point is and write down the coordinates.

(4,1)

2) Plug in the p and q values from the vertex

y=a{(x-p)}^2+q

 

y=a{(x-4)}^2+1

3) Find a point on the graph and use the coordinates to replace x and y.

I’ll use the point : (3, 4)

4=a{(3-4)}^2+1

4) Isolate a

  • 4=a{(3-4)}^2+1
  • 4=a(-1)^2+1
  • 3=a(-1)^2
  • 3 = a

The equation is : y=3(x-4)^2+1

 

Let’s look at how we can determine the domain and range and the intercepts by the quadratic function:

Example: y=3{(x-2)}^2+1

It is always important to first find the vertex as it is the essential piece of information to be able to determine the form of the graph.

y=3{(x-p)}^2+q
  1. Vertex: remember the vertex is (p,q). In our example, p is -2 which becomes +2 because when you place -2 in the standard form, it will become positive with both negatives:
  • y=3{(x-(-2))}^2+q
  • y=3{(x+2)}^2+q

q is the y coordinate of the vertex (not the y-intercept like in general form) which will be 1.

Our vertex is (2, 1)

2. Domain: x∈R – Domain is always the element of real numbers. 

3. Range: we should think about the direction of opening to be able to tell the range. Since the coefficient is +3, the graph opens up. This also means that the vertex is a minimum point with y-coordinate 1.

Therefore, y ≥ 1

4. Direction of opening: Opens up

5. Equation of the axis of symmetry: This is the line that cuts through the vertex.

AOS: x = 2

6. The intercepts:

To find the y-intercept, x=0. Replace x with zero in the equation and solve.

y=3{(0-2)}^2+1

y = 13

To find the x-intercept, y=0. Replace y with zero in the equation and solve.

  • 0=3{(x-2)}^2+1
  • -1=3{(x-2)}^2
  • -\frac{1}{3}={(x-2)}^2

This equation has no solution, so there are no x-intercepts.

 

Inquiry Paragraph: How does it feel to be Transgender?

Posted on by shannonf2015

 

How does it feel to be Transgender?

Image result for lee mokobe

http://cjung-allen.wixsite.com/genderverification/single-post/2013/05/01/LEAVING-SOUTH-EAST-ASIA

In the spoken word “What it feels like to be Transgender” by Lee Mokobe, the author expresses his emotions and feelings as an uncomfortable girl, wanting his desired identity to be a boy. Although his body parts, face structure and voice completed the requirements of a girl, anatomy did not defeat his belief that inside lived a caged boy. As someone who is religious, he continued to pray, hoping Jesus would fix him, but no answer arrived. When his mother tells him, he could be anything he wants, he decides to be a boy. Despite feeling shameful and conflicted with questions, he starts dressing like one, unaware of the consequences he may face with his family: “When I turned 12, the boy phase wasn’t deemed cute anymore. It was met with nostalgic aunts who missed seeing my knees in the shadow of skirts, who reminded me that my kind of attitude would never bring a husband home, that I exist for heterosexual marriage and child-bearing.” (Mokobe). His mother, ashamed of his decision and afraid that he will get hurt or disappear without a trace of words, has put him down for wanting to express himself. This shows that he does not have the support of his family or anyone at his school, and therefore, he is on his own. The environment that surrounds him has also become haunting and harmful as there are many transgenders that lost their life. His life has become a spectacle, everyone flooding him with unwanted questions of curiosity, mixed with ignorance and persistence. He was treated as if he was not human and used as a freakshow on social media: “That my body is a feast for their eyes and hands and once they have fed off my queer, they’ll regurgitate all the parts they did not like” (Mokobe).  As a result of being a transgender, many feel they are doing wrong and feel ashamed of wanting to change genders. They are filled with uncertainty, not many being embraced and supported by their family, and are afraid of the conflicts that may arise. Their life continues to be in the hands of danger and they are confused and lost on how to take action. Therefore, life of a transgender is a self-discovery journey that is filled with obstacles such as shame and conflict as well as unwanted media on their queerness, which may result in family conflicts and becoming the spotlight of danger and social media.

 

 

Works Cited

Mokobe, Lee. What it feels like to be Transgender. May 2015. March 2018. <https://www.ted.com/talks/lee_mokobe_a_powerful_poem_about_what_it_feels_like_to_be_transgender>.

 

Week 7 – Interpreting the Discriminant

Posted on by shannonf2015

Last week, we have learned how to determine the number of solutions of a quadratic equation without solving the equation. To be able to do that, we look at the discriminant!

The discriminant is the radicand of the quadratic formula: b^2-4ac

This is how to determine the number of solutions by the discriminant.

  • Two real roots when b^2-4ac > 0

If the discriminant is positive (so greater than 0), it has two solutions. (It touches the x-axis twice)

  • Exactly one real root when b^2-4ac = 0

If the discriminant is zero, that means there is exactly one solution (Just touches the x-axis once)

  • No real roots when b^2-4ac < 0

If the discriminant is a negative number (so smaller than 0), it has no solutions (Does not touch the x-axis)

Now that we understand how to use the discriminant to determine the number of solutions, let’s apply these to some equations.

First example: Calculate the value of the discriminant and determine how many solutions.

4x^2+2x+3=0

Remember the quadratic equation is : ax^2+bx+c=0

Therefore: a = 4, b = 2, c = 3

Plug in the numbers to determine the discriminant.

b^2-4ac

2^2-4(4)(3)

4-16(3)

4-48

-44

Since the discriminant is negative, it has no solutions; no real roots.

Now let’s look at another example where we have to create an equation with a given number of roots.

Determine the values of y of the equation which has two real roots.

5x^2+3x+y=0

To have two real roots, the value of its discriminant must be greater than 0.

So we use: b^2-4ac > 0

Substitute: a = 5, b = 3, and c = y

3^2-4(5)(y) > 0

Simplify.

9 – 20y > 0

Move the 9 to the other side to isolate the variable.

-20y > -9

Divide both sides by -20.

y > \frac{9}{20}

Since we divided both sides by a negative number, we must switch the inequality symbol.

y < \frac{9}{20}

 

 

Opinion Piece – Should Teams use First Nations as Mascots or Logos?

Posted on by shannonf2015
First Nations for Sports

https://canadianhistoryworkshop.files.wordpress.com/2013/01/treaty-8-logo_fliped.jpg

Sports teams have used First Nations as mascots and logos for many years, causing controversy on whether the act should continue or be prohibited. Some indigenous people find the team logos offensive and unnecessary, while others, including fans of the popular sports teams and First Nations themselves, believe it is not derogatory towards First Nations.

There are positive influences in using First Nation culture as a logo. For one thing, it brings awareness to those who are blinded to notice the presence and history of First Nations. It carries their footprints through a well-known brand, giving them recognition for being the owners of the land. While this may seem like a positive trend, there is still controversy on certain team logos and mascots, which dehumanizes them with exaggerated features or stereotypical images. This demeans the Aboriginal people resulting in them being placed into an inferior category; therefore, there needs to be a limitation. Having said that, referring to them as First Nations does not demean them. For these reasons, sports teams should be allowed to use First Nations as their logo, provided they seek their permission and approval. Take Edmonton Eskimos as an example. The term “Eskimo” referred to the Inuit people as eaters of raw meat; in other words, unintelligent cavemen. The Inuit community have spoken out hoping for a name change, but no change was made. Despite having the federal government and Inuit organizations stop using the word in early 1970s, the Edmonton CFO football team continue to use it. Words used to appropriate a culture and turning them into entertainment is disrespectful. In this case, the Edmonton Eskimos should have considered the Inuit people and communicated with them first to avoid causing harm.
Using First Nations as logos should not be banned, but there are certain boundaries that need to be considered. Rita Pyrillis, a Native American born and raised in Chicago, has shared her story of the difficulty living with transparency through her writing Sorry for not being a Stereotype. People assume she is an immigrant as her appearance does not fit the perception of a First Nation to others surrounding her. They see right through her, unable to tell what is fact or what is fiction: “Sometimes strangers think I’m from another time. They wonder if I live in a teepee or make my own buckskin clothes or have ever hunted buffalo.” (Pyrillis). Based on these stereotypes, they are treated as simple people stuck in the past. To this extent, sports teams should research the history behind the names they plan to use to avoid generalizing an image of First Nations through their logo. In the graphic novel Rising Above by Steven Keewatin Sanderson, he shares examples of positive and negative stereotypes that exists throughout accessible resources. Positive stereotypes deconstruct First Nations, adding new perceptions to their image, which both raises the standards and gives an unrealistic image. Negative stereotypes generalize their actions and appearance, giving an identity that small minded people would understand. Steven, using the example of Pocahontas, wrote about the fact that many illustrated novels sexualize First Nations, portraying them as weak and nothing more: “Aboriginal women are more than just objects to be sexualized, they are strong powerful leaders and the givers of life.” (Sanderson, 39). In view of the above, sports teams should not deconstruct First Nations to meet their standards for logos. They are acceptable as long as they keep their logos realistic, respecting and honouring First Nations true identities.

In conclusion, it is acceptable for sports teams to use First Nations as their logos while being mindful and respectful of the symbolic meanings and significance to the First Nations. Researching about their history and communicating with the owners of the culture is far more acceptable than just borrowing from it.

Works Cited

Obed, Natan. Edmonton Eskimos Name Debate. 22 November 2017. March 2018. <https://globalnews.ca/video/3874931/edmonton-eskimos-name-debate>.

Snowdon, Wallis. Edmonton Eskimos name an insult to the Inuit, says local Inuk woman. 14 November 2017 . Monday March 2018 . <http://www.cbc.ca/news/canada/edmonton/edmonton-eskimos-name-change-inuk-critic-1.4401422 >.

 

Week 6 – Solving Quadratic Equations

Posted on by shannonf2015

This week, we have learned how to solve quadratic equations by completing the square and applying the quadratic formula when the equation is not factorable.

A quadratic equation includes at least one squared variable and has the following form: ax² + bx + c = 0

a, b and c are constants or numerical coefficients and x is the unknown variable.

Let’s start with the method of completing the squares. This method is quite a long process, it depends on the equation. The best would be to see if it is factorable, if it is not or factoring is too difficult for the equation, you can use the completing the squares method or using the quadratic formula.

Completing the Squares 

Lets start with an example:

{x^2}+{4x}=2

To be able to solve this equation, it must equal to zero as it is a quadratic equation. In this example, we have to move the 2 to the other side so that it becomes zero:

{x^2}+{4x}-2=0

It becomes -2 because when you move it to the other side of the equation, you switch the signs.

Now that we have it written in the proper equation for us to solve, first check if it is factorable. As there are no numbers that both multiply to equal to two and add to equal 4 (only option is 2\times{1}). So we must use either the completing the squares or formula. In this example, I will be using the completing the squares method.

First, we make two places that are both + and – to get zero pairs in between the inner term and the constant:

 

Then we find what is the missing constant. To do that, we divide the inner term in two and square it. The answer will be 4. Then we factor for those three terms.

 

Now that you got {(x + 2)}^2, write the other numbers that are next to it: -4 -2 which equals to -6.

Isolate the variable by moving -6 to the other side of the equation. Then square both sides to get x + 2 = \sqrt{6}. Make sure to include both + and – signs next to the radical because it has two options. 

Then move the 2 to the other side of the equation to isolate x.

x = -2 ±\sqrt{6}

 

Now let’s see how we can apply the quadratic formula. 

{2x}^2-2x-1=0

This is the same form as:

{ax}^2+bx+c=0

a=2, b=-2, c=-1

Formula:

Image result for quadratic formula

Substitute the variables with the numbers given.

 

Once you substitute it, simplify it while making sure to keep the + and – sign next to the radical.

 

 

Week 5 – Factoring Polynomials

Posted on by shannonf2015

I haven’t really learned much starting this week, but I have learned quite interesting new techniques on factoring polynomials.

Let’s start off with this acronym I have learned to help with the process.

CDPEU – Can Divers Pee Easily Underwater?

Common

Difference of Squares

Pattern

Easy

Ugly

Let’s breakdown what each term means before getting into the factoring process.

  • Common means find what is common between the terms, the GCF (Greatest common factor).
  • Difference of Squares means if there are two binomials and they are subtracting.
  • Pattern means if they have three terms following the pattern: x^2 x #
  • Easy means the easy ones to factor out (straightforward) : x^2
  • Ugly ones are the polynomials that not so straightforward and need more work. We’ll see what method we can use later on. : a{x^2}

Now let’s apply these steps!

To start, refer to the list above we just talked about.

The first step is find what is common. To do so, write all the possible factors of both coefficients and see what variables both terms have in common.  

 

After we found all possible factors of 49 and 14, we can see that the common number is 7. Both terms also have at least one x variable. Therefore, you would factor out 7x :

Write out the rest of the values in the bracket next to 7x, making sure when you multiply the values in the bracket with 7x, it will give you the same original expression.

Do we stop here? Well let’s check the list.

Difference of squares is the next one. To tell if it’s a difference of squares, it must have two terms and also be able to form conjugates (subtraction of two terms).

In our example, (7x – 2) does not have an x^2, so we cannot factor them out any further. Therefore, it is not a difference of squares.

Next step is pattern. Well this doesn’t work because it doesn’t have three terms, therefore, we cannot simplify it any further.

It is therefore : 7x(7x – 2)

Now let’s look at an example with three terms.

First step is looking for something in common. As we can see, there’s nothing in common so let’s go to the next step.

Second, is it a difference of squares? No! There is three terms.

So, thirdly,  let’s check if there’s the pattern : x^2 x #

Yes!

The first thing to do is draw two brackets and start with putting in x to get x^2. Then factor out all the possibilities for 24.

To decide which two numbers we must use for 24, the two numbers that are multiplying must equal to 10 (the inner term) when added together. So the pair to choose would be 6×4. The sign depends on the inner term. It’s positive and so is 24 so therefore we use ++.

This is the final expression factored.

(x + 4)(x + 6)

To check if it’s correct, we can foil back in the values, meaning distribute all the numbers back by multiplying.

 

So far, these are both easy. But let’s get into the uglier ones.

There are no common factors here, and there are no differences of squares. There is the pattern though with the three terms, but this one seems a little more complicated than the other one we just did.

A method I learned for these ugly trinomials is using the box.

You split a box into four sections, writing in first the value with x^2 which in this example is 25{x^2} on the top left corner and the constant on the bottom right corner.

We multiply both 25{x^2} and 4 which equals to 100{x^2}.

Write out all the possible factors of 100 and find which pair adds together to equal to the inner term.

 

 

When all values are placed in the box, find the greatest common factor in all the values that are next to each other, not diagonally from each other.

 

Final thing that I have learned which for me personally is such a life saver, is being able to replace values with variables.

Here is what I mean.

If we are given this example:

 

 

We can simply replace (3y + 1) with any variable to make the expression more simple.

Now that we have the expression 2ya – 4a, we need to find the greatest common factor, which is 2a.

Then we have to replace the variable back with 3y + 1.

 

The final expression factored is: 2(3y + 1)(y – 2).