# Quadratic Relations

### Amjad Alsabagh

## Expanding

*Polynomials:* Expressions with two numbers or variables added or subtracted together.

**Example: (x+1) +/- (y-2)**

But now we will discuss __MULTIPLYING__ polynomials. Also known as expanding.

**Example: (x+1) * (x+3)**

**x + 3x + 1x + 3**

**x+ 4x + 3**

## Factoring

Factoring a quadratic: working backwards from the quadratic form to find the two polynomials involved

X²+ 2x + 1

X² + Bx + C

1. You find two integer that multiples to C and adds to B

2. after you find the two numbers you put them in (x ) (x ) form

**EXAMPLE**

**X²-5X + 6**

** -2X-3= 6**

** -2+-3= -5**

**(X-2) (X-3)**

**FACTORING A QUADRATIC EQUATION WITH AX² OTHER THAN ONE:**

*4x²+ 4x + 1*

*1. you multiply A with C*

*2. you find a number that multiplies to C and adds to B*

*3. you put the two numbers inplace of B*

*4. you group them and take their factors as the answer*

**EXAMPLE**

**4X²+ 4X + 1**

4X² + 4X + 4

2+2= 4

2X2= 4

4X² + 2X + 2X + 1

(4X² + 2X) + (2X + 1)

2X(2X + 1) 1(2X+1)

__(2X+1)(2X+1)__

## Solving

All you do is take the equation and take the x^2 as A the X as B and the Number as C

Ax^2 + Bx+ c

then you plug it into the formula below

## Completing the square

1. factor out the first number

2.square and divide the 2nd number by 2

3.you then take the negative one and put it on the side of C

4. dont forget to multiply it by the number you have factored before leaving the bracket

5.after that you just factor the equation your left with using the tricks I showed in the "Factoring" section

**Example:**

**2x^2 + 4x + 3**

**2(x^2 + 2x) +3**

**2(x^2+2x + 2) -4 + 2**

**2(x^2 +2x + 2) + 2**

**After this step you can just factor it and you will have the complete square**