Standard Form

Unit 3

Learning Goals:

1.I will be able to find the X intercepts using the Quadratic Formula

2.I am able to complete the square.

3.I can solve revenue, rectangle and triangle problems.

Summary

Completing the Square:

Squaring a Binomial = Perfect Square Trinomial

(a+b)^2 = a^2+2ab-b^2


Example:

(x+3)^2 = x^2+3x+9


How to make it a perfect square, if it's not given:

Consider the following:y=x^2+ 8x+5

1. Put '5' outside of the bracket

y=(x^2+ 8x)+5


2. Divide 'b' by 2 then square it. In this case it's 8.

(8/2)^2 = 16

Then the equation becomes y=(x^2+8x+16-16)+5


3.Move the negative 16 outside the bracket.

y=(x^2+8x+16)-16+5


4.Write as squaring a binomial

y=(x+4)^2-11


Therefore the vertex is (-4,-11)

The Quadratic Formula:

Consider the Following:

All quadratic equations of the form ax^2+bx+c=0 can be solved using the quadratic equation.


This is the quadratic equation:

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The quadratic formula is used to find the x-intercepts of a parabola.

The values for the variables a,b, and c are taken from standard form quadratic equation.


For Example:

Use the quadratic formula to solve for the X-Intercepts

3x^2+5x+2=0


We know that a=3, b=5, and c=2

1. Substitute in the values.

2. Add the vaules together in the square root.

3. Square root the result.

4. Add and subtract the outside number with the product of the square root.

5. Then divide the number you get from subtracting and adding with 2a.

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Now from solving the quadratic formula, you can now plot the the x-intercepts.


Here is a video by mahalodotcom on Youtube on how to solve the quadratic formula.

How to Solve Quadratic Equations

The Discriminant:

Inside the the quadratic formula there is the discriminant.


The equation for the discriminant is:

b^2-4ac


If D>0 there are two x-ints

If D<0 there are no x-ints

If D=0 there is one x-int

Word Problem: Completing the square

Ms. Dhaliwal runs a snowboard rental business that charges $12 per snowboard and averages 36 rentals a day. She discovers that for each %0.50 decrease in price, her business rents out two additional snowboards per day. At what price can Ms. Dhaliwal maximize her profit.


R=(price)(quantity)

let 'x' represent # of decreases


p=(12-0.5x)


q=(36+2x)


R=(12-0.5x)(36+2x)

=432+24x-18x-x^2

=-x^2+6x+432

-1

=-(x^2-6x)+432 (6/2)^2=9

=-(x^2-6x+9-9)+432

=-(x^2-6x+9)+432-9(-1)

=-(x^2-6x+9)+432+9

=-(x-3)^2+441


x=3


p=(12-0.5(3))

p=12-1.5

p=10.5


Therefore, Ms. Dhaliwal should change her price to $10.5 because she can maximize her profit to $441