# Quadratics

## Introduction:

This website will give you an overview of the topic "quadratics". We are infact surrounded by quadtratics in our everyday lives without even knowing. Quadtratics isn't just about parabolas and graphing it is also in fact about factoring, completing squares, word problems, step pattern, quadratic formula, discriminant and much more.

## Table of contents:

Basics

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

## Basics

What is a parabola?

a graph of quadratic relations that can open downward or upwards which means it could either be positive or negative. meaning it could be minimum or maximum

## 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

## Factored form

Y=(x-r)(x-s)

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

Y=ax^2+bx+c

• 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

A represents the value of "a" from the equation. Meaning if the value of a from the formulas is less than 1 the direction of opening would be minimum and if it is more than one it would be maximum

## 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.

## Discriminant

The number in red is the discriminant

## Reflection

I feel that the unit of quadratic so was in fact the hardest and longest unit from my point of view because although at times I felt as if I knew what I was doing I often got confused for what to do. On my test I did not succeed to list the vertext and x intercepts because I was not sure which formula to use since we had learnt so many. However, after creating this website I got a better look at quadratic so and although I still get confused a bit I have a better grasp of the concept. Also, hopefully by looking at others websites I will get an even better idea on quadratic so.