Quadratic Relations Lesson

By: Cindy Cheng

A table of values represents a quadratic relation ie. for constant increments of the independent variable (x), the first differences (△y) are variable and the second differences [△(△y)] of the dependent variable (y) are constant.
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Common Factoring

  • factoring is the opposite of expanding
  • if every term of a polynomial is divisible by the same constant, the constant is called a common factor


Example:

ab+bc=a(b+c)

_ _ _ _ _ _ _ _ _ _> common factor

4x+20=4(x+5)



  • a polynomial is not considered to be completely factored until the greatest common factor (G.C.F) has been factored out



Example:

4x+20=2(2x+10) → not completely factored

4x+20=4(x+5) → completely factored


Try it out:

a) 3x+9

b)12x-16

c)24+16x-8x²

Factoring By Grouping

=x³-2x²+8x+16


bracket off first 2 and last 2 terms


=(x³-2x²)+(8x+16)


factor the brackets


=x²(x+2)+8(x+2)


since (x+2) is the same, you can bring it together


=(x+2)(x²+8)

Factoring by Grouping

Factoring Simple Trinomials

=x²+7x+6


find two numbers that multiplies into 6 and adds to 7


P:6

S:7


the 2 numbers are 6 and 1.


=(x+6)(x+1)

Factoring Complex Trinomials

=2x²+3x-5


nothing that multiplies to -5 and adds to 3. So you have to multiply a(2) with c(-5) which would be your third term


P: -10

S: 3


decompose the middle term


=2x²-2x+5x-5


factor by grouping


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

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

Factoring Difference of squares

=x²-81


if both terms can be squared and the second term is a negative, it can easily be factored.


=(x+9)(x-9)

solving quadratic equations by factoring

x²+3x+2=0


factor off the equation


(x+1)(x+2)=0


there will be 2 answers that equal x in the end.


x+1=0 and x+2=0

x=-1 and x=-2

completing the square

=3x²+24x-11


block off the first 2 terms


=(3x²+24x)-11


factor out the A


=3(x²+8x)-11


add "zero" inside the brackets. To get zero-- (middle term/2)²


=3(x²+8x+16-16)-11


bring out the negative


=3(x²+8x+16)-11-48

=3(x²+8x+16)-59


factor the rest


=3(x+4)²-59

Completing the Square - Solving Quadratic Equations

quadratic formula

formula you need to remember:




x2 + 3x – 4 = 0


a=2

b=3

c=-4


plug into the formula and answer. you will get 2 answers.

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Using the Quadratic Formula

The End