Quadratic Relations

Amjad Alsabagh

Topics

1.Expanding

2.Factoring

3.Solving

4.Completing the square

Expanding

First we will start off by understanding what polynomials are.

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


When factoring a quadratic equation with AX² of 1:

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

In order to solve a quadratic you have to use the quadratic formula


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

Big image

Completing the square

To complete the square here is what you have to do

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