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