# Quadratic Systems

### Dinesh's Weird Problems With Parabolas

## Table of Contents

2.Introduction to Parabolas

3.Steps of Solving Parabolic Problems

4. Expanding and Simplifying

5.Example of Parabolic Problems

6.Understanding The Use of Parabolas

7.Conclusion And Other Information Links

8.Part Two Factoring Quadratic Equations

9.Introduction to Factoring

10.Types of Factoring

11.Polynomial Factoring

12.Simple and Complex Trinomial Factoring

13.Word Problems With Factoring

14.Conclusion and Other Informational Links

15.Part Three Maxima and Minima

16.Introduction to Maxima and Minima

17.Introduction to Solving Quadratic Equations

18.Solving Equations

19.Solving Using The Quadratic Formula

20.How To Find The Vertex

21.Wrap-Up

22.Informational Sites On Quadratics

## Introduction to Quadratics

## Introduction to Parabolas

## Steps of Solving Parabolic Problems

1. Graphing the line.

Graphing parabolas is the best way to solve all systems involving lines. In quadratics to graph the parabola first you need to create a table of value chart. When your done the second step is to add the x and y variables onto the chart. The third step is to multiply the x numbers on the chart by its self twice because the x in the equation is squared so when you do that, the answer you get is the y intercepts.

## Expanding and Simplifying Quadratic Equations

What will you do when you have a factored form like this? 2(x+3) (x+4). Basically what you would do is leave the 2 outside the brackets and just multiply the first to terms to the second brackets. You will have 2(x2 + 4x + 3x + 12) now you will add the two like terms in the brackets. After you have done that you will have 2(x2 + 7x + 12) the actual answer is what you have to do here. The whole point of this is to expand and simplify so just multiply everything in the bracket by 2. The equation will be 2x2 + 14x + 24.

## Example of Parabolic Word Problems

**An object is launched at**

**19.6**

**meters per second (m/s) from a**

**58.8**

**-meter tall platform. The equation for the object's height**

**s****at time**

**t****seconds after launch is**

**s****(**

**t****) = –4.9**

**t****2**

**+ 19.6**

**t****+ 58.8**

**, where**

**is in meters. When does the object strike the ground ?**

2. **A picture has a height that is ****4/3**** its width. It is to be enlarged to have an area of ****192**** square inches. What will be the dimensions of the enlargement?**

3. A basketball has been shot. The ball was shot 10m high and was 5m low before being scored. How high was the ball before it reached the basket?

These are word problems that are solved by graphing, factoring, solving for x-inters, and using the vertex form and find vertex.

## Conclusion and Informational Links on Parabolas

## Part Two: Factoring Quadratic Equations

## - Introduction to Factoring

It is basically isolating a quadratic equation to find two numbers for a quadratic problem. For example

Expanding (x+3) (x-3) into x2 - 9 and Factoring x2 - 9 into (x+3) (x-3).

How many ways are there to factor?

1. Common Factoring

2. Factoring Simple Trinomials

3. Factoring Complex Trinomials

4. Factoring By Grouping

5. Differences of Squares

6. Perfect Squares

## 1. Common Factoring

For Example: x2-2x-8 factor find the two factors of c that gives you c when multiplied and b when adding. two numbers 2,-4. into (x+2) (x-4)

## 2. Factoring Simple Trinomials

For example: x2+2x+6 there is no number infront of the x2 or no a value.

## 3.Factoring Complex Trinomials

For example: 2x2-10x+12 then find GCF of all terms 2(x2+5-6) and find the two numbers that give you c value when multiplied and b value one adding 2(x-2) (x-3)

## 4. Factoring By Grouping

For example:

xy-5y-2x+10 into xy-5y -2x+10 then you find GCF of it y(x-5) -2(x-5) Then take the factors and the x-5 in both expressions and put it into one expression. (x-5) (y-2)

## 5. Differences of Squares

For Example: you have a2-b2 you have to make them into ( ) ( ) so what you do is add the two A's into the two brackets and the two B's into two brackets. (a,b) (a,b) in order for you to have a negative expression a2-b2 you have to add in one bracket and subtract in another. (a+b) (a-b)

## 6. Perfect Squares

For example: x2+10x+25 find the two numbers that gives you b and c well guess what you get the same number which is 5. Why 5x5= 25 and 5+5=10 so in this case the answer will be (x+5)2 because 5 gives you both the numbers 10 and 25 and you don't have to write (x+5 (x+5) instead this is a short cut to that way of writing the answer.

## Part Three: Maxima and Minima

## 1.Finding The C Values

## 2. Solving Equations

## 3. Quadratic Formula

so in this case your equation is ax2+bx+c=0 and all you do is add the values for each variable from the equation. Sometimes you can use this to check if your previous way of finding the x-intercepts and see if you get the same x-intercepts in this too.

## 4. Vertex Form: To Find Vertex of The Parabola

*y*=

*a*(

*x*–

*h*)2 +

*k*. Lets say you have an equation like this y= x2 + 4x - 5 basically what you have to do is put the brackets between the x2 and 4x so it will be something like this. Know you will have y= (x2 + 4x) - 5 now what you do is find the c value which is 4/2 and equals 2squared so it would be y= (x2+4x+2squared - 2squared ) -5 and then it would be y= (x2+4x+4) -4 -5. Now finally it would be y= (x+4)2 -9 and your vertex will be (4,-9). Now you can graph the parabola. If have a question like why do you have to square it to two? Here is the reason why... When we were finding c we were doing that for the c value outside of brackets, but here we are dividing a square root so we would have to have the number squared.