QUADRATICS
Learn more about quadratics
Multiplying Binomials
-> When muliplying binomials always remember FOIL
F- stands for first
O- stands for outer
I- stands for inner
L- stands for last
Example: (x+2) (x+3)
1st step: (x+2) (x+3) "x" muitiplied by "x" = x squared
(First)
2nd step: (x+2) (x+3) "x" multiplied by 3 = 3x
(outer)
3rd step: (x+2) (x+3) 2 multiplied by "x"= 2x
(inner)
4th step: (x+2) (x+3) 2 mulitlplied by 3 = 6
(last)
Rewrite: xsquared + 3x + 2x + 6
collect like terms: xsquared + 5x + 6
^ this now is a simple trinomial
* keep reading down and learn how to solve simple trinomials*
Common Factoring
NOTE: Common factoring is the opposite of expanding
-> An example is given below which step by step shows you how to solve for common factoring.
Ex: 10x+5
1st: find GCF for 10 and 5
10- 1,2,5 and 10
5- 1 and 5
GCF: 5
2nd: Write soultion with brackets but first divide GCF number 5 with 10x+5
10x/5 + 5/5
- GCF goes on the outside of brackets
5(2x+1)
Lastly check answer
5(2x+1) = 10x+5
Factor Simple Quadratic Expressions of the Form x^ + bx + c
NOTE: When factoring a polynomial of the form of x^ + bx + c (when a=1) we should find the:
- two numbers that add to give b
- two numbers that multiply to give c
-> to help you understand this expression completely always write x^ + bx + c before solving
Here is an example: x^ + 5x + 6
x^ + bx + c
- before doing anything find the sum and product 2 multiplied by 3 = 6 / 2 plus 3 = 6
Therefore the answer is (x+2) (x+3)
Factor complex quadratic expressions of the form ax^ + bx + c
-Here is an example on how to solve for complex trinomials
-There will also be a video after the example, if you still haven't fully understood the concepts to complex trinomials
Example: 3x^ + 8x + 5
ax^ + bx + c <- rewrite the formula
3x^ + 8x + 5 <- multiply the "a" with "c"
product: 3 times 5 = 15
sum: 3+5=8
-Factor by grouping:
(3x^ + 3x) (5x + 5)
= 3x(x+1) + 5(x+1)
- now write (x+1) (3x+5) <- this is the final answer for now
NOTE: The video below shows another method, you can chose whatever method suits you they both give you the same answers.
Factor a perfect square trinomial
-Here is easy example of a perfect square
- Afterwards watch a video that is a bit more complex step by step
Example: Expand and simplify this: (x +4)^
(x + 4)^
=(x+4) (x+4)
- Before doing anything remember FOIL
-Use FOIL to simplify
First: x time x
Outer: x times 4
Inner: 4 times x
Last: 4 times 4
= x^ + 4x + 4x + 16
- Now collect like terms which are (4x,4x) add these terms together
= x^ + 8x +16
Factor differences of square
Example: 9k^-25=0 (differences of squares)
(3k+5) (3k-5)=0
(3k+5)=0
3k divide 3 and 5 divide 3
k=5 over 3
(3k-5)=0
3k divide 3 and -5 divide 3
k= -5 over 3
maxima and minima
- The difference between max and a min is that a max will have an negative in front of the "a" value and the min would be positive which means that it can go further up
Example of a Maxima: y=-x^ + 8x - 3
Example of Minima: y=x^ + 4x - 3
Transformations
-label each property, so it's easy to know what is what
- here is an example below
- draw chart for organization
PROPERTIES/ Y (X+3)^
-vertex/ (-3,0)
-A.O.S/ x=-3
-stretch or compression/ none
-direction of opening/ upward
-value x may take/ set of real numbers
-Value of y may take/ y=-3
word problems
Now its getting harder, now there is a word problem and you have to solve it by using any quadratic equation or formula.
- remember highlighting or underling important things in word problems helps a lot
The word problem is:
graphing parabolas in y=a(x-h)^+k form
- I will be teaching how to find the y and x vertex by taking turns and letting x=0 and y=0
- also then teaching how you can use the square root or quadratic formula to solve
- then graphing the parabola
the quadratic formula
- quadratic equations that can be factored are really easy to solve but those quadratic equations that cannot be solved there is a formula for them
Formula:
x= -b-+ square root (b^-4ac)
divided by 2a
-Sorry guys I did do my video but its taking forever for it to upload so I couldn't post it