# Quadratic Relationships

### By: Tavia Martin

## What Is Quadratics & How Is It Useful

## QUADRATICS IN THE REAL WORLD

## Table Of Contents

**Types of equations**

- Factored form

- Standard form

- Vertex form

**Factored form**

- Factors and zeroes

- Expanding (factored form to standard form)

- Common factoring

- Factoring simple trinomials

- Factoring complex trinomials

- Special cases

- Difference of squares
- Perfect square

- Factoring by grouping

- Graphing factored form

**Vertex form**

- Isolating for x

- Graphing vertex form

**Standard form**

- Solving using quadratic equation

- Discriminant

- Completing the square (standard to vertex)

- Graphing from standard form

__Word problems__

## FACTORED FORM

__Factors and Zeros__

Factors: A factor is one of the linear expressions of a single-variable polynomial. A polynomial can have several factors, such as the factors...

(x - 1) and (x + 3).

The factors of a polynomial are important to find because they can be multiplied together to gain a polynomial.

When polynomials are graphed, many of them intersect the x-axis. The locations where a polynomial crosses the x-axis are called ‘zeros.’

Zeros: A zero is the location where a polynomial intersects the x-axis. These locations are called zeros because the y-values of these locations are always equal to zero.

When polynomials are graphed, many of them intersect the x-axis. The locations where a polynomial crosses the x-axis are called ‘zeros.’ ,

## Expanding (Factored Form to Standard)

** Expanded Notation: **Writing a number to show the value of each digit. It is shown as a sum of each digit multiplied by its matching place value (units, tens, hundreds, etc.)

The factored form of a quadratic is useful for finding the roots and other properties but sometimes it is useful to expand in order to simplify the equation.

Expanding from factored form uses the distributive property of real numbers which states for real numbers x and y,

c (x + y) = cx + cy

** Standard Notation:** Just the number as we normally write it. Example: 382 is in Standard Notation.

## Common Factoring

**Factors are the numbers you multiply together to get another number**

__Common Factors:__When you find the factors of two or more numbers, and then find some factors are the same ("common"), then they are the "common factors".

Example: 12 and 30

• The factors of 12 are: 1, 2, 3, 4, 6 and 12

• The factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30

So the common factors of 12 and 30 are: 1, 2, 3 and 6

## Factoring Simple Trinomials

## Factoring Complex Trinomials

## Difference of squares

## Perfect Squares

## Factoring By Grouping

## Graphing Factored Form

**Step 1**:Decide if the four terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.

**Step 2**:Create smaller groups within the problem, usually done by grouping the first two terms together and the last two terms together.

**Step 3**:Factor out the GCF from each of the two groups. In the second group, you have a choice of factoring out a positive or negative number. To determine whether you should factor out a positive or negative number, you need to look at the signs before the second and fourth terms. If the two signs are the same (both positive or both negative) you need to factor out a positive number. If the two signs are different, you must factor out a negative number.

**Step 4**: If the factors inside of the parenthesis are exactly the same, it is time for the 2 for 1 special. The one thing that the two groups have in common should be what is in parenthesis, so you can factor out what is inside the parenthesis, but only write what is inside the parenthesis once. If what is inside the parenthesis does not match, you need to rearrange the four terms and try again until you get a perfect match. If you have rearranged the problems a couple of times and still have not found a perfect match, then the problem does not factor.

**Step 5**: Determine if the remaining factors can be factored any further.

## VERTEX FORM

## Isolating for 'x'

step by step:

1. **Combine like terms.**

2. **Simplify the equation.**

3. **Move terms with X's in them to the left side of the equation.**

4. **Move all parts of the equation except the terms that have an x in them to the the right side of the equation.**

5. **Divide both sides of the equation by the coefficient.**

6. **Solve for x.**

## Graphing Vertex Form

## STANDARD FORM

## Solving Using Quadratic Equation

## Discriminant

In algebra, the **discriminant** of a polynomal is a function of its coefficients, It gives information about the nature of its roots.

## Graphing in Standard Form

## Word Problems

## TYPES OF EQUATIONS

- · Factored form :
**a(x-r)(x-s)** - · Vertex form:
**a(x-h)²+k** - · Standard form:
**ax²+bx+c**