What is Quadratics and how is it useful?

The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2).

It is also called an "Equation of degree 2" (because of the "2" on the x). Quadratic are visually represented in a graph by U shaped lines called Parabolas. A Parabolas is a curve where any point has an equal distance (except for the optimal value). Thinking about it, you may be asking yourself, "how is this useful?". Quadratics is very useful in all levels of study from business to science. If you want to find the maximum revenue a company can make, or to see how high you can throw a football before it lands, you would use quadratics.

Table of Content

Terms and definitions

Part 1

Equations/formulas

vertex form

factor form

optimal values

Part 2

Formulas

Multiplying Polynomials

Factoring

Factoring by grouping

Factoring simple trinomials

Factoring Complex trinomials

Perfect squares

difference of squares

Part 3

Formulas

Completing the square

Finding points

Discriminants

Word problems

Economic problem

Area and perimeter problem

motion problem

Maximize the Area of a fence with the given perimeter

Reflection

Part 1

Unlike linear relationships, quadratic relationships have different first differences and the same second differences. If you see a table that looks similar to this one, know that it will be graphed in a parabola.

Formulas

Vertex form:y=a(x-h)^2+6

factor form: y=a(x-r)(x-s)

determining the equation given the Vertex

Often times you are asked to determine the equation of a parabola given the vertex and a point that the graph passes through. Although this might sound confusing, this is how it is done.

Vertex form

y=a(x-h)^2=k. This equation is called vertex form because it is very easy to find the vertex. The x and y coordinates of the vertex are literally in the equation. Each letter in the equation determines the characteristics of the graph (look at terms and definitions to see what each does) .An example of vertex form is y=5(x-4)^2=6. The graph below shows what the graph will look like.

factor form

Factor is slightly different from vertex form. Factor form is when you are given 2 points on the parabola that cross the x axis. The equation looks like this: y=a(x-r)(x-s). Like vertex form, factor form has an "a" value that (also like the vertex form) determines weather the graph will stretch or compress. For this graph, the equation would look like: y=(x-5)(x-1)

solving Formulas

a^2+2ab+b^2--> (a+b)^2

a^2-2ab+b^2-->(a-b)^2

(a^2-b^2)-->(a+b)(a-b)

Multiplying Polynomial

1. To multiply polynomials, you need to multiply each term in one polynomial by each term in the other polynomial. Add those answers together, and simplify if needed. Often times you are given bigger equations that needs to be expanded and simplified, for example: 2(3x+5)(2x-6)-6(3x-6)(4x+5). Although it may look tantalizing, the same steps are required to solve these bigger equations (also make sure to use BEDMAS).

Factoring polynomials

Factoring polynomials isn't quite the same as factoring regular numbers. When factoring polynomials, you have to find numbers and variables that divide into the polynomials. when factoring, you have to keep what's common outside the bracket, and what is not, inside. Also, make sure that the common factor is common in all the polynomials. In the picture, the common factor is 2x.

Factoring by grouping

Factoring by grouping is splitting a long expression up into two parts, finding a common factor for each of the two parts, and then finding a common factor for the whole expression. This will make more sense when it is shown with an example:

factoring simple trinomials

If you remember back to standard form, the equation looks like this; (ax^2+bx+c). In simple trinomials, "a" is equal to 1 (hence simple trinomials).
factoring simple polynomials is similar to part 1 where we talked about factor form. You have to see what two numbers multiply to get "c" and add to get "bx". Factoring simple trinomials is will look like this.

factoring complex trinomials

Okay, factoring simple trinomials was pretty simple. Lets move on to something more complex...Like complex trinomials. The major difference between simple and complex trinomioals is the "a" value. In simple trinomials, "a" is equal to 1, but in complex trinomals, "a" is not equal to 1. When solving for complex trinomials, you have to multiply the "a" and the "c" value. When you get the product of the two values, you have to find two factors of the same product, but also add to "bx" value as well. If you're unsure on whether or not your answer is correct, you can always check for answer. If you forgot [or have no idea as to] what I am talking about in the context to the values, this is the expression I am referring to: ax+bx+c
Factoring Simple and Complex trinomials

perfect squares

when faced with a problem like(x^2-8x+16 , you cant factor it by grouping, or finding common factors. You have to solve these equation by using squares. This is what is called perfect squares. When given an expression that needs to factored using squares, the express will look like this: a^2+2ab+b^2. To determine if a expression can be solved with perfect squares, you have to see if the "a" and "b" values (values from each side) can be square rooted like in the example on the right. When you square root "a" and "b" you have to divide the middle number by two and 2 and see what 2 factors make up that product. After that, you have to simplify.

example for perfect squares

My explanation might sounded a little confusing, so hopefully this example will help.

difference of squares

when solving for difference of squares you have to turn (a^2-b^2) into (a+b)(a-b). If you look at the first equation you might notice the subtraction sign. That sign tells us the two numbers will be cancelled out. Here's an example: (x^2-100)---> (x-10)(x+10). When we try to expand (x-10)(x+10), you're left with x^2-10x+10x-100, as you can see, the two numbers in the middle cancel each other out.

Formulas

ax^2+bx+c--> y=a(x-h)^2+k

-b/2a

Completing the square

When working with squares, we have primarily worked with the formula ax^2+bx+c. If we take a trip down memory lane, you might recall the original formula we have been working with: a(x-h)^2+k. In Completing a square, we try to go from ax^2+bx+c -->a(x-h)^2+k. Completing a square can be a little difficult, so I've prepared a video explaining how it is done.

Factor Form

When solving a quadratic equation, your goal is to find the x-intercepts of the equation. When given an equation that is in factor form, it is very easy to solve. Example: (x+3)(x+4)=0 (remember to find the x-intercepts, you have to set the y value to zero). All you have to do is rearange the equation so that x is isolated. x=-3 & x=-4

Vertex

If you want to be able to find the vertex of 2x^2+10+14 what you have to do is turn the equation into standard form by completing the square. After completing the square, your equation should look like this:2(x+5)^2-39. You can find the vertex by looking at the h,k values. The vertex for this equation is -5 and -39.If you want to find the aos, you can use the formula -b/2a by just plugging in the numbers and solving.With the aos, you can find the optimal value just by plugging in the aos value (x) into the formula and solving.

Just like factoring, completing squares and graphing, the quadratic formula is just another way used to find the solution for a quadratic equation. All you have to is plug in the values into the equation and solve. If you're wondering which values go where, all you have to do is look at the equation ax^2+bx+c. Example: 5x^2+3x+7. a=5 b=3 c=7.

You might be confused by the +and- sign next to the "b" value. What you do is add, and subtract the product so that you have to answers (2 x-intercepts).

Discriminants

Discriminants is the expression under the square root that tells you the nature of the x-intercepts (b^2-4ac). There are 3 things that need to be noted about Discriminants: 1 when D<0 (d means Discriminants) then there is no x-intercepts, 2 when d=0 then there is 1 x-intercept, 3 when d>0 there is 2 x-intercepts

Word problems

Now that we've covered the basics, the only thing left to do is apply those newly found skills to word problems!

Area and Perimeter problem

The area of a rectangle is 105cm^2. The length of the rectangle is 5cm more than the width. What are the measurements of each side?

Motion problem

Luffy the pirate has a cannon that’s underwater. The height of the cannonball above water level t second after it was fired is given by h=- 4.9t^2 +20.6t-8.

a) How far under water is the cannon?

b)how long will the cannon be above water?

c) what is the maximum height the cannon ball will travel.

Maximize the Area of a fence with the given perimeter

You have a 500-meter roll of fencing and a large field which is boarded on one side by the building. You want to construct a rectangular playground area. What are the dimensions of the largest yard. What is the largest area

Reflection

Test Reflection

This was the Quadratics Mini test 1. I performed poorly on this test but I am aware of my mistakes and have improved. This test mainly covered everything in Part 1 of this flyer.

Unit reflection

Quadratics was a very useful unit, and it is very obvious how it can applied in a variety of real-life application. It can be used for maximizing the revenue of a business, maximizing the area of a fence with a given perimeter, and is great for many other applications as well. The Quadratic formula was also a very useful and important formula when trying to find the x-intercepts. The basics of grade 10 quadratics will be pursued in the further years of our academic lives, so it is very important that we understand the basics now. If you need help with understanding quadratics, here is a links that might be of some assistance