# Quadratics

### Review package

## Quadratic Relations

- One variable is related to another by a function involving constant terms and terms of 1st order or higher.
- A polynomial with a degree of 2 is Quadratic Relation.
- For constant increments of the independent variable, a relation is quadratic if the second differences of the dependent variable are constant.
- A graph that is either concave upwards or concave downwards
- When the rate of change is constant it is a linear relation
- When the rate of change is not constant then its not a linear relation

## FACTORING

- Factoring is finding what to multiply to get an expression
- To factor a number means breaking it into an equation where a number can be multiplied to get the original

There are many different types of factoring I will explain all.

## Common Factoring

Firstly we have - __Common Factoring__

- If all the terms in the equation are divisible by the same number or variable then that term is called the
*common factor* - A polynomial is not completely factored if the GCF (greatest common factor) is not factored out.
- Expanding and factoring are 2 completely different things
*a(b+c) <= (factoring) ab+bc <= (Expanding)*

__Examples__- 25
*x*+50

*x*+2)

=> (factored Fully because 25 is the GCF)

*2. *25*x*+50

=> 5(5*x*+10)

=> (It is NOT factored fully as 5 and 10 still have a common factor)

## Grouping

factoring by grouping is factoring in done by grouping pairs of terms. Then factor each pair by groups with ()

__Examples__

x^2+2x^2+8x+16

=> x^2(x+2) +8(x+2)

## SIMPLE TRINOMIALS

- Simple trinomial equations are in the form of, x^2+bx+c
- In solving trinomials, the brackets have to be expanded (Multiply each term in the first bracket to each term in the second bracket).
- Equations in the form of x^2+bx+c are called
*Quadratic Equations*

Expand=> x^2+bx+c , (x+2) (x+1)

- (x+2) (x+1)

= x^2+3x+2 - "c" value is the product. Find out 2 numbers that add up to the "b" value and when multiplied the answer is the "c" value.

__Examples:__- x^2 +7x -30 [Product=-30 , Sum=7]

= (x+10) (x-3)

## COMPLEX TRINOMIALS

__COMPLEX TRINOMIALS__

- Factor by Decomposition.
- Complex Trinomial are in the form of, Ax^2 +Bx +C.
- The "C" value is multiplied by the A value, only the coefficient.
- Determine the factors of the product which also add up to the "B" value.

Factor by Decomposition: Ax^2 +Bx +C

- 2x^2 +3x -5 [Product: A*C= -5*2=-10 , Sum=3]

=2x^2 +5x-2x -5 [Decompose the Middle term]

=x(2x+5) -1(2x+5)

=(x-1) (2x+5)

__Examples:__- 5x^2 +10x+2x+4

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

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

## FACTORING DIFFERENCE OF SQUARES

__FACTORING DIFFERENCE OF SQUARES__

- The "a" value is squared (x^2)
- In the form of, (a+b) (a-b)
- Square-root of "b" value

Expand:

- (x+2) (x-2)

=x^2 - 4

- x^2 -16

= (x^2 - 4)

__Example:__4x^2 - 22

=2(2x^2 - 11)

## COMPLETING THE SQUARE

- Used to change Standard form equations to Vertex form

STEPS TO COMPLETE THE SQUARES:

- Block off the first two terms
- Factor out the A value only
- Divide the middle term by 2 and square it (b/2)^2
- Add and subtract the squared value answer
- Take out the negative squared value, Multiply the "a" value by the negative squared value when taking it out of the bracket
- Write within brackets x + the square root of the number and square the entire bracket and solve what is outside the brackets

__Example:__- x^2 + 6x - 2

=(x^2 + 6x) - 2

=(x^2 + 6x + 9 - 9) -2

=(x^2 + 6x + 9) -9 - 2

=(x+3)^2 - 11

## QUADRATIC FORMULA

- In the form of, ax^2 + bx + c = 0
- Formula is used to identify the "x" value
- Quadratic formula is used when the equation cannot be factored
- Quadratic formula give 2 values of "x"
- Quadratic Formula:
x = -b ± √b^2 - 4ac /2a

__EXAMPLES:__- x^2 - x + 8 = 0

x= -b ± √b^2 - 4ac /2a

x= 4x^2 + 33x + 8 = 0

x= -33 ± √33^2 - 4(4)(8) / 2(4)

x= -33 ± √961 / 8

x= -33 ± 31 / 8

x= -1/4 x= -8

THE END HOPE IT HELPED!!!!!!!!!!!!!!

Using the Quadratic Formula

Completing the Square - Solving Quadratic Equations

Factoring difference of squares