# Quadratics For Beginners

### Learn all about the grade 10 math curriculum!

## Introduction

Quadratic relationships are different than linear relationships as they are curved line instead of straight ones. Also unlike linear relationships, quadratic relationships can have 2 x-intercepts. In this website, you, a math enthusiast, will learn about different quadratic equations, the forms they may take and how to convert and solve them.

## Different Quadratic Equation Forms

__Vertex Form:__

**y= a(x-h)*2+k**

*This allows you to easily identify the vertex (h,k)*

__Factored form:__

**y= a(x-r)(x-s)**

*This allows you easily identify and solve for the x-intercepts*

__Standard form:__

** y= ax*2+bx+c**

*This allows you to solve using the quadratic formula*

## Let's Get Started On Chapter 4

__In this chapter you will find:__

-First Differences

-Vertex form & transformations

-Graphing vertex form** {y=a(x-h)*2+k}**

-Graphing factored Form** {y=****a(x-r)(x-s)}**

-Word Problems

## First Differences

In a quadratic relationship, the first differences are not equal but the second differences are. In this relationship the second differences are all equal to 2.

When the first differences and second differences do not all have the same number, then the relationship is neither linear or quadratic.

## Vertex Form & Transformations

If the variable "a" is positive, then the parabola (the curved line) opens upwards, and so if the same variable is negative, the parabola opens downwards.

The variable "a" also controls the transformations of the parabola.

If "a" is less than one, the parabola is thinner.

If "a" is more than one, the parabola is wider.

If "h" is positive, then the parabola will translate to the left.

If "h" is negative, then the parabola will translate to the right.

If "k" is positive, then the parabola will move up.

If "k" is negative then the parabola will move down.

## Graphing Vertex Form

Secondly you must identify the step pattern (Up_ Over_).

Then with this knowledge you will be able to plot the vertex, and two other points in this graph thus making a parabola.

Below is a wonderful video explaining graphing from vertex form by the amazing Mr.Anusic and Mr.James.

## Graphing Factored Form

This form gives you the x-intercepts of a parabola.

To graph this, you have to add your "r" and "s" values and then divide by 2.

This gives you the value of x, allowing you to substitute it and solve for your y coordinate.

## Word Problems

A bridge has an arch which is represented by the relationship y=-2(x-2)(x-4). x represents the distance horizontally from the start of the bridge in meters and y represents the height of the arch in meters.

a) what length of the bridge does the arch cover?

b) what is the height of the bridge at 3m in distance?

## Let's Get Started On Chapter 5

__In this chapter you will find:__

-Expanding polynomials

-Expanding special products

-Common factoring binomials

-Factoring by grouping

-Factoring simple trinomials

-Factoring complex trinomials

-Factoring a perfect square

-Difference of square

## Expanding Polynomials

## Expanding Special Products

1. (a+b)*2

2. (a-b)*2

3. (a+b)(a-b)

1. (a+b)*2

To expand this, you can use Sam Doesn't Pull Strings.

S - Square the first term

DP - Double their product

S - Square the second term

To expand this, you can use the Sam Doesn't Pull Strings method as well!

To solve this, you just multiply the first two terms, and the last two, and **DO NOT** multiply the inner and outer terms as they cancel each other out.

If you have difficulty with this, just spread the rainbows and simplify.

## Common Factoring Binomials

## Factoring By Grouping

You then factor each group, and factor again as the numbers and variables in the brackets should be the same.

This gives you your final answer.

## Factoring Simple Trinomials

A simple trinomial is an expression in standard form and the first coefficient starts with the number 1** {x*2+bx+c}**

You substitute "bx" for those two factors and then factor by grouping, as taught above.

## Factoring Complex Trinomials

A Complex trinomial is an expression in standard form and the first coefficient starts with the number greater than one 1 {ax*2+bx+c} (The sign does not matter, as long as the number is anything other than one, it is a complex trinomial).

In order to factor a complex trinomial, you must first common factor if it is possible.Secondly, you find factors of "a" that multiply with factors of "c" that add up to give you "b".

## Factoring Perfect Squares

The first and last terms must be able to be square rooted, and the middle term must be the product of those roots, times two.

Do do this, you can use the Sam Doesn't Pull Strings method.

## Difference Of Squares

Both terms should have a square root.

You may use the Sam Doesn't Pull Strings method here, but you only need to use the Sam and Strings part.

## Let's Get Started On Chapter 6

__In this chapter you will find:__

-Finding zeros in vertex form

-Completing the square (Standard form to vertex form)

-The quadratic formula

-Optimization word problems

## Finding Zeros In Vertex Form

**{**y= a(x-h)*2+k}

In order to find zeros in vertex form, you must substitute "x" for 0 and then solve for "y."

After doing this, you can substitute the value of "y" into the equation and solve for the value(s) of "x."

NOTE: when finding the value(s) of "X" you might end up with no value, one value, or two values, as a parabola can intercept the x axis these many times.

When square rooting a number, you can get 2 zeros as the square root of 16 is 4 and -4.

Make sure you branch off your equation into two at this point and continue solving!

## Completing The Square (Standard Form to Vertex Form)

Firstly you have to add and subtract half of "b*2" to your equation.

You then factor the equation, without the last two terms and simplify.

This then gives you the same equation in vertex form!

## The Quadratic Formula

However this formula can only be used to solve equations in standard form.

All you have to do is substitute values into the formula and solve!

NOTE: Don't forget that square rooting a number will give you two answer, one will be negative and the other will be positive. This gives you the value of the x-intercept(s) if there are any.

you can give an exact answer, which is leaving the question in formula form, or an approximate answer, which is solving for the value of x, sometimes resulting in a decimal answer.

## Optimization Word Problem

Attached below is a video demonstrating how to solve these problems!