# Quadratics

### By Jasman S.

## Introduction:

## Table of contents:

Vertex form

Step pattern

Parts of a parabola

Factored form

Standard form

Converting standard form to vertex form

How 3 form relate

Factoring

Expand and simplify

Common factoring

Factoring simple trinomials

Factoring complex trinomials

Factoring perfect square trinomials

Factoring differences in squares

Quadratic formula

Discriminant

Word problem

Reflection

## Vertex Form

a(x – h)2 + k

In order to know the vertex form you need to know the basic steps:

THE FORMULA

THE STEP PATTERN

HOW TO GRAPH IT

PROPER TERMINOLOGY

## Step pattern

If you take the "standard" parabola, y = x², which has it's vertex at the origin (0, 0), then:

Starting from the vertex (0,0) as "the first point" ...

OVER 1 (right or left) from the vertex point, UP 1² = 1 from the vertex point

OVER 2 (right or left) from the vertex point, UP 2² = 4 from the vertex point

## Parts of a parabola

## Factored form

The form of an algebraic expression in which no part of the expression can be made simpler by pulling out a common factor.

In factored form, there are two x intercepts (s&t)

EX: y=-2(x+3)(x+1)

x intercepts @ (-3,0)(-1,0)

(-3+-1)/2

Therefore; the axis of symmetry is -2.

## Standard form

a, b and c are known values. a can't be 0.

"x" is the variable or unknown (we don't know it yet

a, b and c are known values. a can't be 0.

"x" is the variable or unknown (we don't know it yet

a, b and c are known values. a can't be 0.

"x" is the variable or unknown (we don't know it yet

Standard form cannot be graphed therefore we convert standard form to vertex form

## Converting standard form to vertex form

Since we cannot graph standered form. how do we change it to vertex form?

We complete a perfect square!

## Direction of opening

## Factoring:

## Expand and Simplify:

__To expand we get rid of the brackets.__

Distributive law:

a(b + c) = ab + ac

We use the distributive law to write expressions such as 5(x + 4) in a form that has no brackets,

5(x + 4)

= 5x + 20

## Common factoring

You know how to multiply two brackets together. (expand and simplify)

(a + b)(c + d) = ac + ad + bc + bd

If you have the expression on the right hand side how can you change it back to the two brackets on the left hand side?

When you common factor, you are breaking it up by finding a number or expression/exponant that is common

Example: (2x+6)

there is a common factor of 2

=2(x+2)

Example 2:

## Factoring Simple Trinomials x^2+bx+c

Factor:

Example 1.

1)x^2+6x+5

= (x+1)(x+5)

Solve: =x^2+5x+1x+5

x^2+6x+5

Example 2:

x^2+10x+16

=(x+8)(x+2)

Solve= x^2+2x+8x+16

=x^2+10x+16

Example 3:

x^2+7x+12

=(x+3)(x+4)

Solve: x^2+4x+3x+12

=x^2+7x+12

## Factoring Complex Trinomials ax^2+bx+c

A __complex trinomial __is a trinomial that does NOT start with 1.

For example: 3x^2+18x+12

2x^2+11x+15

Look for a common factor, and factor it into 2 sections

Break into 2 factors; guess and check

=(2x+5)(x+3)

you need to find 2 numbers that multiply to 15.

Put the first two terms in brackets and then the last two terms in brackets

Then factor

Finally , put the terms outside the brackets in one separate bracket, and the terms already in brackets combined into one bracket

## Factoring perfect square trinomials

__Perfect squares__ start with a squared number and end with a squared number with a number in between (a^2+2ab+b^2)

Example 1:

2x^+8x+4

=(2x+2)(x+2)

then expand to check

2x(2x^2+4x)^2+(2x+4)

=2x^+8x+4

## Factoring differences of squares

__Difference of squares __have nothing in the middle (x^2-36)

Example: x^2-36

=(x+6)(x-6) because the square root of 36 is 6 and when the equation is expanded it goes back to original form indicating that it is the right factors.