# Quadratics

### By: Mohit Devassar

## Table of content

**Parabolas**

- Step Pattern

- First differences

- Second differences

- Vertex Form

- Optimal value

- X-intercepts/zeros

- Transformations

**Factoring**

- Trinomial Factoring

- Binomial factoring

- Algebra Tiles

- Factoring by grouping

- Factor form

- Perfect squares

- Square differences

- Standard form to factored form

**Solving**

- Standard form to vertex form

- Quadratic formula

- What is the quadratic formula

- Discriminant

- Solving from vertex form

- Solving from standard form

- Word problem

- Reflection

## Bridges A bridge is an example of a man made parabola in our world. | ## Rainbow Rainbow is an example of natural parabola created by mother nature. | ## Dolphin This Dolphin is jumping out of the water and back in which looks like a parabola |

## Parabolas vertex form

## Step Pattern

## Graphing parabola in vertex form

## First Differences

## Second Differences

## Vertex Form equation

## Vertex Form

## Axis of symmetry

## Optimal Value

## How to graph with Vertex form

## X - intercepts/zeros

Example:

y = 2(x+2)^2-8

0 = 2(x+2)^2-8

8 = 2(x+2)^2

8/2 = (x+2)^2

4 = (x+2)^2

+-2 = x+2

-2 = x+2

-2-2 = x

-4 = x

## Transformations

The A value controls if the parabola open up or down.

The H value controls the horizontal movement.

The K value controls the vertical movement.

*All of these values must be used when using transformations.*

If the A value is greater then 1 then the parabola is being stretched and if it is smaller then 1 then the parabola is being compressed and if it is 0 then nothing changes.

If the K value is increased then it would move up by that many numbers and if it is decreased then it would move down by that many numbers.

If the H value is increased then the it would move left by that many numbers but if it is decreased then it move right by that many numbers

## FACTORING

## Algebra Tiles

## Factoring Binomials

= 2(x+2)

I got this by finding the greatest common factor (GCF) and putting it to the side and dividing everything by the GCF which got me what was left in the bracket. You can check if your answer is right by distributing the 2 back in the bracket. It should equal back to the original equation.

Check:

2(x+2)

= 2x+4

## Factoring trinomials (complex) (simple)

*Complex:*

2x+4x+16x

= 2x(2+8)

*Simple:*

6x+15

=3(2x+5)

The process is the exact same as factoring Binomials but there is just three numbers that is why it makes it trinomials. A simple equation does not have a a value. (x+4x+16x) but you would factor it the same way.

## Factoring by grouping (simple) (complex)

*Complex:*

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

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

*Simple:*

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

= (x+4) (x+2)

All you have to do is find something common like (x+3) and (x+3) and make them represented by one binomial instead of two and then just group your remaining numbers in this case it was (2x+5) and then make it into a factored form equation by putting them beside each other like in the example.

## Factoring standard form

## Factored Form y=a(x+a)(x+b)

## Factoring Perfect squares

A perfect square is any trinomial that follows a^2+2ab+b. Another way the perfect square equation can look is ax^2+b^2. If this is the form, make sure all coefficients are square rooted. An example to this would be turning 9x^2+81 into 3x^2+9^2.

## Differences of Squares a2-b2

## Standard Form to Factored Form

## Factored form to Standard form

## Solving

## Standard form to vertex form

## Quadratic formula

## What is the quadratic formula

## Discriminant

The Discriminant is the number inside the square root of the equation.

1) If the Discriminant is negative then there is no solution.2) If the discriminant is 0 then there is 1 solution.

3) If the Discriminant is positive then there are 2 solutions.

## Solving from vertex form

## Solving from standard form

## Word problem (example)

a) what are the dimensions of the sides?

b) what is the maximum area?