Quadratics

Everything you need to know to be an expert!

Introduction

In this presentation, we will review on simple knowledge of quadratic equations and the different ways to solve them.


Definition-

Quadratic Equation: A quadratic equation is any equation which is written in the following form, ax²+bx+c=o

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Topics:

1) Factoring

2) Expanding

3) Completing the Square

4) Quadratic Formula

Factoring

Factorization is the first way to solving a quadratic equation, in where you decompose the equation as a product of multiple factors. In order to assure that you have factored correctly, after you multiply the factored equation, it should give you the original quadratic

Factoring Quadratics

In order to factorize an equation, you need to replace the original numbers with their factors.

Example: Original Quadratic- x²+5x+6
Factored Form- (x+3)(x+2)
As you can see if we multiple this factored form, if it is correct it's product will be the original equation.
Let's try another equation, but this time in steps
Original Quadratic: x²+2x-15
Step 1- When you first look at the equation, remember that when your finding your factors c will be the product, and b will be the sum
Step 2- Once you find your two numbers which multiply to -15 and add up to +2, which in this case are -5 and 3, re-write your equation so it multiplies out to the original equation
Factored Form (x-5)(x+3)

Factoring By Common Factor:
In some cases a quadratic equation is different and meant to be factored by common factor, which is as simple as it sounds. You have to find a common factor throughout all the numbers and divide a, b, and c by the common factor
Example: 2x³+22x²+56x
Factored Form: 2x(x²+11x+28)


Solving By Factoring:
After you have factored the quadratic equation and have put it into factored for, the next step which is to find out what x is, you have to solve for x as the bracket is equal to 0.
Basic Example: x²+7x+10=0
(X+2)(X+5)=0
x+2=0 or x+5=0
x+-2 or x=-5
Advanced Example: 8x²+33x+4=0 P:32 S:33
8X²+1X+32X+4=0
x(8X+1)+4(8x+1)=0
(8x+1)(x+4)=0
8x+1=0 or x+4=0
x=-⅛ or x=-4
Solving Quadratic Equations by Factoring.avi

Expanding

Expanding is basically the opposite of factoring, once you are given an equation in factored form, to expand you multiply it all out

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Examples:
1) ( x + 3 ) ( x + 2 )
= x ( x + 2 ) + 3 ( x + 2 )
= x²+ 2x + 3x + 6
= x² + 5x + 6

2) (x+10) (x+8)
=X (X+10) (X+8)
=X²+10X+8x+80
=x²+18x+80

Completing The Square ❑

Completing the square is another way of solving a quadratic equation, I will show you a step by step process of how to complete the square on any quadratic equation

y=x²+8x-3
Step 1) Block off the first 2 terms
y=(x²+8x)-3
Step 2) Factor out the A
Step 3) Divide the middle term by 2, then square it
8/2²=16
y=(x²+8x+16-16)-3
Step 4) Take the negative number out
y=(x²+8x+16)-16-3
y=(X+4)²-19

Solve for X

Example:
y=(x+2)²-9=0
√(x+2)²=√19
x+2=±3
x=±3-2
x=1,-5
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Quadratic Formula

Lastly, the most convenient way of solving a quadratic equation is by using the Quadratic Formula

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Example:

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Here is a video that demonstrates a couple of examples that apply the quadratic formula

Using the Quadratic Formula

Word Problems:

Examples:

1) The product of two consecutive negative integers is 506. What are the numbers?
Topic: Factoring
Process:
n
(n + 1) = 506
n2 + n = 506
n2 + n – 506= 0
P:506
S:1
(n +23 )(n –22 )=0
n=-23, n=22
Answer: Therefore, since the question is asking for a negative integer, the first number would be -22 and its consecutive integer which is -23


2) The height of a bungee jumper in the air t seconds after he jumps can be modeled by the function y= t² + 10t +9, where t and y are measured in feet. What is the minimum height that the jumper reaches?
Topic: Completing The Square
Process:
y=t²+10t+9
y=(t²+10t)+9
y=(t²+10t+25-25)+-30
y=(t²+10t+25)-25-30
y=(t²+10t)-5
y=(t+5)²-5
Vertex:(-5,5)
Answer: Therefore, the minimum height the jumper reaches is 5 ft.

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