Quadratics

By: Dannia Kamakh

Properties Of A Quadratic

  • The quadratic relation is called a parabola
  • The vertex of the parabola is either the greatest point on the parabola or the smallest point
  • If the parabola opens up it has a minimum
  • If the parable opens down it has a maximum
  • If the information is in a table then the second differences determines wether it will open up or down if the second difference is negative then it opens down if its positive it opens up
  • The vertex's co-ordinates are (h,k) x=h and y=k
  • The equation of the axis of symmetry is x=h
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Multiplying Binomials

Using the distributive property you expand then simplify.

For example:

(x+3)(x+3)

You first multiply the x in the first bracket with the x in the 2nd bracket and the number in the 2nd bracket then you repeat the process only using the 2nd number in the first bracket this time.

=x^2+3x+3x+9 *** The first number should always have a ^2 and there should also be 2 other x's***

Then you simplify and get:

x^2+6x+9

Common Factoring

This is the opposite of expanding

ab+ac= Expanding

a(b+c)= FactoringFor example: 10a+5a+40

10a+5a+40

You find a common factor between the first 2 numbers or 3 and divide every term by it.

=5(5a+a+8)

Factoring By Grouping

When you are able to factor it completely until you reach factored form.

x^3 +2x^2+8x+16

so you first common factor it

=x^2(x+2)+8(x+2)

Then you find what else they have in common (x+2) and factor it. And put whatever is in-between in brackets in the 2nd pair of brackets .

=(x+2)(x^2+8)

Factoring Simple Trinomials

We have to use the sum and product rule where you basically take the first and last number and multiply them, and the product of those 2 numbers is the sum of the middle number.

For example:

x^2+7x+6

So 6 x 1=6 (Now I have the product) now I need to figure out what 2 numbers add to +7 and multiply to 6?

6 and 1 do (6 x 1=6 and 6 + 1=7)

so i simply put them into the brackets with the x's

(x+6)(x+1)

If I wanted to check and expand and simplify I should get the original equation back.

(x+6)(x+1)

x^2+6x+x+6

x^2+7x+6

Factoring Complex Trinomials

Its basically the same thing as Simple Factoring but there is a number in front of the x so its not worth 1 anymore as a result your product is a bigger number. For Example:

8x^2+22x+15

P=120 S=-22

=8x^2+10x+12x+15 (sub in the 2 numbers that multiply to get the P and add to get the S)

=2x(4x+5)+3(4x+5) (Factor by Grouping)

=(4x+5)(2x+3) (Factored Form)

Completing the Square

You block off the first 2 terms and take the ''a'' out (the first number but keep the variable in the bracket) , you then divide the middle term by 2 and square it and make sure it equals to "0" then take the negative number out of the brackets to get vertex form. For Example:

y=x^2+6x+11

y=(x^2+6x)+11 (Block of first 2 terms)

y=(x^2+6x)+11 (Factor)

y=(x^2+6x+9-9)+11 (Divide middle number by 2 and square it, make it equal to "0")

y=(x^2+6x+9)+11-9 (Factor inside the bracket)

y=(x+3)^2+2 (Vertex Form)

The Quadratic Formula

This formula is very simple all you do is lane you first number a, your second number b and third number c and plug it in to this equation it will quickly give you your x's (zero's) and you will be able to go from there For Example
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Graphing and Solving Quadratic Equations

Equation: y=-x^2 -2x+3

1. You need to factor it by completing the square

y=-x^2 -2x+3

y=-x^2-3x+x+3

y=-x(x+3)-1(x+3)

y=(x-1)(x+3)

2. You now have your factored form quadratic equation no we can find the x and y intercepts

y=(x+1)(x+3) (To get your x-int you need to make each bracket =0)

x=1 x=-3

to get you y-int you need to make your x's equal to 0

y=(0-1)(0+3)

y=(-1)(3)

y=3

3. To find your vertex you need to get your x's add them together and divide them by 2 this will give you your h

h=(3)+(-1)=2

h=-2/2

h=-1

Now you take your h and plug it back into the original equation where the x's are

y=-x^2 -2x+3

y=-(-1)^2 -2(-1)+3

k=4

vertex=(-1,4)

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Your Turn!

Graph and Solve:

x^2+4x+3

Answer: x=-1,-3 y-int=3 vertex:(-2,-1)

and

2(x+2)^2

Answer: vertex:(-2,0)