# Quadratic Exploration

### By: Kesigan

## What is quadratics and how does it apply to everyday life?

The word Quadratic comes from "quad" meaning square, because the variable gets squared. The graph of a quadratic which is a parabola. Quadratics can be used to find the flight of objects. When throwing a ball the path the ball creates is a parabola. Parabolas can help find: the highest point reached by the ball, how long it takes to reach the highest point, what is the initial height of the ball, etc. Other real life scenarios can be used to calculate the parabola of roller-coasters or suspension bridges. Furthermore, quadratics are important in the business world. Quadratics inform companies about how many customers to expect, how to increase their cost, etc. In addition to business, quadratics increase our understanding dimensions of shapes. It allows us to find the area and perimeter to new shapes. By going through this website you will be a quadratic master!

## Table Of Contents

- Introduction to vertex form
- Graphing by step pattern
- Word problems using vertex

Section 2: Factored form equations

- Expanding and simplifying
- Monomial Factoring (GCF)
- Binomial Factoring (GCF)
- Factoring by grouping (4 terms)
- Simple trinomial factoring
- Complex trinomial factoring
- Special Product- Difference of Squares
- Special Product- Perfect Square Trinomial
- One Application problem

Section 3: Standard form and quadratic formula

- Quadratic formula
- Standard form equation
- Completing the square
- Discriminant
- Application word problems

## Learning goals of section 1

The learning goals for section 1 include:

- I can graph
- I know the features of a parabola
- I can solve word problems using vertex form
- I can create mapping notation equation

## Features of the Parabola

- Axis of symmetry: Is a imaginary line that divides the parabola into halves equally. The x co-ordinate of the vertex.
- Vertex: Part of the parabola where it changes direction and is the minimum/maximum point of the parabola. The axis of symmetry pierces through the vertex.
- X-intercept(s)/Root(S)/zero(es): The point where the parabola passes through the x-intercept. The could be up to two x-intercepts.
- Y-intercept: The point where the parabola passes through the y-intercept.

## Two videos on Step pattern graphing

## Chapter 2: Word Problems

y=-3(x-9)²+21.5

1. What is the maximum height of the rocket?

2. How long does it take for the rocket to reach maximum height?

3. What is the initial height of the rocket?

4. How long is the rocket in the air?

5. What was the height of the rocket after 3 seconds?

## The solutions:

2.The time taken to reach maximum hieght is 9 seconds. To find time to reach maximum hieght we just need to find the x co-ordinate of the vertex. Therefore, the time taken to reach maximum hieght is 9 seconds.

3. Initial hieght means hieght when time is equal to 0.

y=-3(0-9)²+21.5

y=-3(-9)²+21.5

y=-3(81)+21.5

y=-243+21.5

y=-221.5

Therefore, the initial hieght is -221.5 meters.

4. We need to find how long the ball was in the air hence we need to set the hieght to 0.

0=-3(x-9)²+21.5

-21.5=-3(x-9)²

-21.5/-3=-3(x-9)²/-3

7.166=(x-9)²

*√*-7.166=*√*(x-9)²

+-2.676=x-9

x=11.68

5. To find the hieght after 3 seconds we need to replace x with 3 seconds to find hieght.

y=-3(3-9)²+21.5

y=-3(-6)²+21.5

y=-3(36)+21.5

y=-86.5

Therefore, the hieght of the rocket after 3 seconds is -86.5 meters.

## Mapping notation video

## Learning Goals of section 2

- I can expanding and simplifying
- I can monomial Factoring (GCF)
- I can binomial Factoring (GCF)
- I can factoring by grouping (4 terms)
- I can simple trinomial factoring
- I can complex trinomial factoring
- I can solve special Product- Difference of Squares
- I can solve special Product- Perfect Square Trinomial
- I can solve one Application problem
- I can graph factored form equations
- I understand factored form equation
- I can find the vertex and axis of symmetry in factored form

## Factored form equation

## What does the a value represent? The a value tells you if the parabola will be stretched or compressed. If there is no value in front of a that means the the parabola is in the shape of a base graph. | ## What does the r value represent? The r value is a x-intercept value. The opposite charge of the r value is the x-intercept. The mathematical way to find the x-intercept is to set the bracket to 0 and solve for x. | ## What does the s value represent? The s value is a x-intercept value. The opposite charge of the s value is the x-intercept. The mathematical way to find the x-intercept is to set the bracket to 0 and solve for x. |

## What does the a value represent?

## What does the r value represent?

## Different types of factoring tutorial

Expanding and simplifying

Simplifying is to put an mathematical terms to the simplest form.

Here I will show you expanding and simplifying:

(x+3)(x+5)

=x²+5x+3x+15

=x²+8x+15

First I used the expansion method which is multiplying both of the terms in the first bracket to the other two terms. The "x²" came from (x)(x), the "5x" came from (x)(5), the "3x" came from (3)(x), and the "15" came from (3)(5). The 3x and 5x were common so, I added them together to create 8x which is called simplifying.

Another way a question might we asked is like this:

(x+9)²

=(x+9)(x+9)

=x²+9x+9x+81

=x²+18x+81

The "²" represents there are two copies of the question therefore, on my first step I wrote (x+9) twice. From here i just expanded the question and simplified.

The question could be asked like this:

3(x+7)(x-4)

=3(x²-4x+7x-28)

=3x²-12x+21x-84

=3x²+9x-84

I used the same expanding method and multiplied 3 around the whole expanded equation.

Monomial factoring:

In monomial factoring we factor out the GCF(greatest common factor) as shown below.

2x+2y

=2(x+y)

3s²+5s²

=s²(3+5)

Binomial factoring:

In binomial factoring, we put the common factor in brackets. The two remaining terms are also put in brackets.

4x(x+5)-2(x+5)

=(4x-2)(x+5)

2x(x+2)-7(x+2)

=(2x-7)(x+2)

Factoring by grouping:

To factor by grouping we just remove a common factor out of the equation.

10x²+5x+4x+2

5x(2x+1)+2(2x+1)

=(5x+2)(2x+1)

In the brackets we have the same expression so we write that once in our final answer. Then the two remaining terms outside the bracket are written in brackets with the other final answer and put them to together as shown above.

Simple trinomial factoring:

Simple trinomial factoring have an a value of 1.

x²+7+10

=(x+2)(x+5)

We need to find two numbers that multiply into 10 and those two same numbers add to 7. These two numbers go into the answer.

Complex trinomial factoring:

Complex trinomials have an "a" value higher then 1.

3x²+5x+2

3x²+3x+2x+2

3x(x+1)+2(x+1)

=(3x+2)(x+1)

The method i used to solve this complex trinomial is to decompose it. To do the compose method you multiply the a value and c value. After we need to find two numbers (same numbers) that when multiplied into 6 and add up to 5. The 5 came from the b value which once again the two numbers must add up to. The two numbers were 3 and 2.

3x2=6

3+2=5

We rewrite 5x as 3x and 2x as shown above. Then you just factor by grouping and now you have solved a complex trinomial!

Special products- Difference of squares:

Differences of squares can be defined as two squares being subtracted. By looking at these types of question we can factor them. Difference of squares formula is this: a²-b²

x²-16

=(x+4)(x-4)

We first factored the x and used a negative 4 and positive 4 to cancel a b value and equal negative 16. With the c value being negative one number in the bracket will be positive and the other will be negative.

A=1 B=0 C=-16

(x+4)(x-4)

=x²-4x+4x-16

=x²-16

=x²-4²

Special products- Perfect square trinomials:

To factor a perfect square trinomial we need to use the decompose method with breaks down the b value so we can later on factor by grouping.

To find out if your equation is a perfect square trinomial we can put it into this formula:

a²+2ab+b² or a²-2ab+b² depending on your b value. If your b value is negative then you use the negative 2 formula otherwise, you use the positive 2 formula.

Example: x²+6x+9

The b value(+6x) is positive hence we use the positive 2 formula. (x)²+2(x)(3)+(3)²

If the b value of the formula equals the original equation then that means it is a perfect square trinomial. The b value of the original expression is 6x and the formula tells us 2xXx3 which equals 6x.

One application problem:

The area of the rectangle is x²+15+56, find the length and width.

To find the length and width you just factor the expression.

x²+15+56

8x7=56

8+7=15

=(x+8)(x+7)

Therefore, the length and width of the rectangle are (x+8) and (x+7).

If the value of x=3, what is the area of the rectangle?

(3+8)(3+7)

(11)(10)

=110cm²

Therefore, the area of the of the rectangle is 110cm².

## How to graph a factored form equation

## Chapter 3

## Standard Form

## What does the a value mean? The a value decides the if the parabola will be stretched or compressed. | ## What does the b value mean? The b value determines the placement of the vertex. | ## What does the c value mean? The c value is the y-intercept of the parabola. The c value will always be a number and can never have a variable with it. It also affects where the vertex will be. |

## What does the a value mean?

## Quadratic Formula

Example: y=3x²-8x+2

x=8+/-/(8²-4(3)(2))/2(3)

x=8+/-/(64-24)/6

x=8+/-6.3

x=2.4

x=0.3

Therefore, this equation has two x-intercepts at (2.4,0) and (0.3,0)

## Completing the square

Steps required to complete the square:

First, put the first two terms in brackets and factor out the a coefficient in the entire bracket.

Next, you divide the new b value by 2 and then square it.

Write a positive value of the number in the bracket and add a negative value of the number outside the bracket. We do this because we can not create new numbers out of no where so the value added is subtracted.

Now, multiply the negative b number with the coefficient in front of the bracket.

After, factor everything in the bracket into a perfect square trinomial and add the numbers outside the bracket.

Finally, you have completed the square and have converted standard form to vertex form.

## Discriminant

Example:2²-4(2)(3)=-20

If the value of the discriminant is 0 then there is one solution to the equation.

Example:4²-4(1)(4)=0

If the value of the discriminant is above 0 then there are two solutions to the equation.

Example:4²-4(1)(2)=8

## Graphing

## Word problems using the quadratic formula

Let x represent the length.

(x)(x-4)=240

Here I have the let statement explaining what the x value represents. I made this equation because lxw=area, hence the formula above is the same.

x²-4x=240

x²-4x-240=

a=1

b=-4

c=-240

x=4+-//-4²-4(1)(-240)/2(1)

x=4+-//16+960/2

x=4+31.24/2 x=4-31.24/2

x=17.62 x=-13.62

I used the quadratic formula to find the x-intercepts. As you can see we have two x-intercepts and we use the positive x-intercept the find the dimensions.

(17.62)(17.62-4)=280

(17.62)(13.62)=280

Therefore, the length is 17.62m and the width is 13.62m.

## Word problems by completing the square

Let x represent

Let y represent

y=(12-0.5x)(36+2x)

y=432+24x-18x-1x²

y=-x²+6x+432

y=(-x²+6x)+432

y=-(x²-6x)+432

y=-(x²-6x+9)-9+432

y=-(x²-6x+9)+9+432

y=-(x-3)²+441

Let r represent the maximum revenue.

r=(12-0.5(3))

r=12-1.5

r=10.5

Therefore, if Jacob sells at $1.50 less then he can maximize his revenue.

## Reflection

## Connections

A factored form equation include the x-intercepts of the equation and tells you the opening of the parabola which helps in graphing. Once you expand the factored form equation, you get the standard form equation which tells you what the y-intercept is.

How vertex form connects to graphing:

Vertex form identifies the vertex of the equation. This point can also tell you if it is the highest or lowest point in your graph.

How standard form connects to graphing:

Standard form tells you what the y-intercept is and standard form can easily be converted to factored form or vertex form. You can factor to find the x-intercepts and furthermore, you can complete the square to find the vertex of the parabola.

## Quadratic unit's assessments

## End of the website

Thank you for learning from our website!

## Works cited

*YouTube*. YouTube, 2013. Web. 09 May 2016.

MathMeeting. "Solve Quadratic Equations Using Quadratic Formula."*YouTube*. YouTube, 2011. Web. 09 May 2016.

PatrickJMT. "Completing the Square Example 2 Solve Quad. Equations."*YouTube*. YouTube, 2008. Web. 09 May 2016.