Quadratics

Impress the teachers and know Quadratics before your taught!

Intro To Quadratics

What is Quadratics:

-The word quadratics come from "quad" meaning square, becuase the variable is getting squared.

-An quadratic equation must have the variable being squared

What are Parabolas:

Parabolas are:


Parabolas are formed by quadratic equations. Just like a line is created by the equation y= mx + b, simply like that, a parabola is created by the quadratic equation.


"a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side. The path of a projectile under the influence of gravity ideally follows a curve of this shape."



Standard Form and Vertex Form:

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Two X- Intercepts:

The two x- intercepts of a parabola are where the parabola touches the x axis. If you look at an example of a ball, the two x intercepts could be seen as a starting point and an ending/ landing point. The two intercepts can help find a lot, such as the vertex.
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Different Quadratics Problems:

How to get vertex from two x-intercepts:

Given the two x- intercepts, you simply add the two of them and divide by two:


- (x-int) + (x-int) / 2 = vertex

Finding the equation with vertex and intercept given:

When finding the equation while you have the vertex, you would have to solve for a and plug in the vertex information in the equation:


Y= a(x-h)^2 + k

The h is the x value in the vertex with the opposite sign, and the k is the y value in the vertex.


Effects that numbers have on Parabolas:


In the equation y= a(x-h)^2 + k:


· a affects the vertical strength

· H effects the horizontal translation (Left/ Right)

· k affects the vertical translation (Up/ Down)

· if a value if negative, there is a reflection

ex y= -1/2 (x – 2)^2 – 5


Transformations:


Ø Reflects the x-axis

Ø Vertical compression of ½ or 2

Ø Shifts two to the right horizontally

Ø Shifts five units down vertical shift

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*Equation & Vertex of Parabola: given the two x intercepts and the y intercept

Factored Form of Quadratic Relation:

If y= a(x-r) (x-s) than:


- Zeros are found by setting each “factor” equal to 0 so:

- x-r = 0 means x=r and x-s = 0 means x=s

- The axis of symmetry is the midpoint of the two zeros so:

- x= r+s/2


- The optimal value is found by subbing the axis of symmetry value into the equation

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Quadratic Relations of the Form y = a(x r)(x s)

Expanding Binomials:

Expanding binomials is just multiplying a number or bracket into the other bracket:


- 2(3-x) = 6 – 2x

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

= 2x^2+ 2x – 3x- 3

=2x^2 – x – 3

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Expanding Binomials

Factoring Polynomials (Common Factors):

When factoring polynomials, we are trying to find numbers that divide out the original polynomials evenly. It just taking out a number that is factorable to everything in the equation. Example:


(2x + 4)= (2x+2*2) – you can take out the two from the equation because the two 2s are common and your equation would be =2(x+2) which is the same as (2x+2*2)

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Factoring trinomials with a common factor

The Main Ways to Solve an Quadratic Equation

There are three types of polynomials:

- simple trinomials

- complex trinomials

- special factoring cases

Factoring simple trinomials is the basics. It involves two brackets to be expanded easily:

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

= 2x^2+ 2x – 3x- 3

=2x^2 – x – 3

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Quadratic Formula:

The quadratic formula is the last way to factor an equation. This way is used when none of those techniques can be used and when the equation is hard to answer such as decimals.


The quadratic formula is:


y= -b +_ square root of (b^2 – 4(a)(c))/ 2(a)

ex. 0= 3x^2 + 4x -15

- a= 3x^2

- b= 4x

- c= -1

5

The quadratic formula has many techniques to it as well where we can end up with one solution, two solutions and no solution.

- When the discriminant ( the answer of the square root) is positive, than there are two answers

- When the discriminant is 0, there is only one answer

- When the discriminant is negative, there are no real roots so there is no solution.

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Using the Quadratic Formula

Quadratic Word Problems:

Read The Question Twice, and Understand it First becuase the word problems can be very confusing

Application Problem with Quadratic Formula

Don't Lose your ball next time, use Quadratcis to find out where you ball went! :P

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Factoring trinomials with a leading 1 coefficient

Factoring complex trinomials involves thinking more :

- a complex trinomial can be turned to a simple trinomial by common factoring as well:

- 3x^2 – 6x + 9, in this we can common factor out the three and make it into a simple trinomial

- In the second case, we would have to try trial and error:

- 2x^2 + 11x + 15

= (2x + 5) (2x + 3)

- You can double check by expanding the equation to see if its correct

Another way we can use is the guess and check table where we have three columns and we put numbers in to see what the two factors will be.
Factoring Complex Trinomials
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The two special factoring cases are perfect squares and difference of squares ad two methods are:

1. (a+b)^2


= a^2 + 2ab + b^2

2. (a + b) (a – b)

= a^2 – b^2


Perfect squares mean that the last number and the first number can be squared perfectly meaning into whole numbers. This also means that the factors will be the same when factored ex. (2x+3) (2x+3)= (2x+3)^2 it can even be two negative numbers ex (x-2) (x-2)= (x-2)^2


Difference of squares is almost the same thing but for this we are subtracting the two squares meaning one of the signs in the brackets will be a negative. One will be positive and one will be negative so it cannot be squared. (2x-3) (2x+3) and (x+2) (x-2)

Completing the Square:

Completing the Square is almost the same as perfect square because they both involve squares. There are just a few minor steps before that and it ends in a different way.


- completing the square is just turning the equation from standard form to vertex form:

- ax + bx + c = 0 -> a ( x+h )^2 + k = 0


Steps:


- First put brakctes around the ax + bx

- check for common factors

- use the method ( b/2 ) ^2 with b

- than take out the fourth number

- use the perfect square or difference of square method to solve

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Completing the square

Factoring Everything:

- Common factoring: 2x^3 – 4x = 2x(x-2)

- Simple trinomials: x^2 + 8x + 12= (x+6) (x+2)

- Difference of Squares: 64y^2 – 81= (8y-9) (8y+9)

- Perfect Squares: x^2 + 4x + 4= (x-2)^2

- Grouping: 3x^2 + 6x + 2x^2 + 4x = (3x^2 + 6x) (2x^2 + 4x) than solve..

- Complex Trinomials: 2x^2 + 11x + 15 = (2x + 5) (2x + 3)