# Learning Quadratics

### Step By Step Instructions

## Linear, Non-linear & Quadratic Relations

*Linear*- a single straight line drawn through all of the point
- first differences are same

__Non-Linear__

- a single smooth curve can be drawn through every point
- the first differences are not the same

__Quadratic__

- first differences are not same but, second differences are
- has a degree of 2

## Forms

There are mainly 3 forms:

- Vertex form
- Standard form
- Factored form

## Multiplying Binomials

A binomial is an expression consisting of two terms which is separated by a positive or negative sign. To multiply two binomials we use the F.O.I.L method as seen below.

## Foil Method

The

**FOIL method**is a way to multiply two binomials.**F**irst- multiply the first terms from each binomial

**O**uter- multiply the outer terms from each binomial

**I**nner- multiply the inner terms from each binomial

**L**ast- multiply the last terms from each binomials

## Common Factoring

- Factoring is the opposite of expanding (multiplying)
- If every term of a polynomial is divisible by the same constant, the constant is called a common factor

## Factor By Grouping

- the first step is to group the first two terms together, and the last two terms together
- factor the terms
- check if you can see a common factor that can be factored out
- put the left over variable and number together as a term

## Factoring Simple Trinomials

- a trinomial is a polynomial with 3 terms
- there has to be a degree of 2
- usually in standard form: ax² + bx+c
- expand using FOIL

- identify a,b,c
- write down all factor pairs of c
- identify which factor pair from the previous step sums up to b
- Substitute factor pairs into two binomials

## Factoring Complex Form

- In the form: ax² + bx+c

__Factoring By Decomposition __

- Multiply the lead coefficient by the constant term.
- Find two numbers that multiply to make the product from step 1, but add to make the middle term coefficient.
- Rewrite the original trinomial, replacing the middle term with two terms whose coefficients are the numbers from step 2.
- Common factor the first two terms from step 3. Then, common factor the last two. Do the pairs separately; it won’t be the same common factor for the first two as for the last two.
- Notice from Step 4 that, although the common factors you took out front don’t match, the brackets do match. Put the common factors in their own bracket, then rewrite

How to Factor (Decomposition)

## Difference Of Squares

## Solving Quadratic By Factoring

The first step to solving polynomial equations is to set the given equation equal to zero. The next step is to factor; and the final step is to set each of the resulting factors equal to zero and solve each equation, which yields more than one solution. Some problems involve factoring a trinomial into two binomials, and some problems involve factoring a binomial as the difference of two squares.

## Completing The Squares

Equation must be in standard form to use this method. This then leads the equation to vertex form.

__Steps:__

- bracket of the first two terms
- Factor inside the bracket but, don't factor out the variable
- You will need to make a perfect Zero so: divide the 2nd term by 2 and square it to get the number
- add the number and the negative f it in the bracket
- move the negative number outside and multiply it by the number in front of the bracket
- then take the number and add it with the number outside the bracket
- inside the bracket take the variable and subtract it by the 2nd term divided by 2
- then square the bracket. so far: (x-number)² + 2nd number
- Now that it is in vertex form you can use it to solve for x or finding the maximum, minimum and more...

## Quadratic Formula

To use this formula it is best to have the equation in standard, to identify a, b & c. This formula is an easy and quick way to find the variable especially for fractions and decimals.

## Changing Forms

This diagram below shows how to change the forms