Standard Form

By: Jasmeen Mann

Learning Goals

1. What is factoring?

2. What is completing the square?

3. The process to change a standard form to vertex form.

4. Know when the points are maximum or minimum.

5. Formula for quadratic relations of the form.

Standard form to vertex form

y=ax^2+bx+c which is standard form

y= a(x-h)^2+k which is vertex form

Completing the Square to turn to Vertex Form

Completing a square means to make an equation a perfect square trinomial. In order to do so, there are a few simple steps to follow:

Step of completing a square

First, put the first 2 terms(x^2 and ) into parentheses. Remember that c goes outside the parentheses. Next, take out a greatest common factor if possible. In the example to the right, a greatest common factor of -9 can be taken out. Keep in mind don't count.



Example: -9x^2+36+8


=(-9x^2-36)+8

=-9(x^2+4)+8

Now divide your value of b by 2 and then square your answer. Whatever the answer you get, add it to the equation, and also subtract it so that the equation is not changed.


-9(x^2+4x)+8

4/2=2

2*2

4


-9(x^2+4x+4-4)+8

Now take out the negative number that is inside the brackets, by multiplying it by the greatest common factor. Outside of the brackets, you may need to add or subtract some numbers. so do all that needs to be done before moving on. In this case, we need to add 36 and 8 outside the brackets.


-9(x^2+4x+4-4)+8

=-9(x^2+4x+4)+36+8

=-9(x^2+4x+4)+44

The last step to do is factoring all that is inside the brackets. it is a perfect square trinomial.


-9(x^2+4x+4)+44

=-9(x+2)^2+44

The final answer should be in vertex form.

-9(x+2)^2+44

Using the Quadratic Formula

The Quadratic formula is used when a cettain quardratic cannot be factored. This formula was developed by completing the square and solving the qudratic ax^2+bx+c=0.


The formula tells us that x=

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Number of Solutions and the Discriminant

Discriminant (D): (b^2-4ac)

When D<0, then their would be no solutions and no x-intercepts.

When D>0, then their would be two soultions.

When D=0, then their would be only one soultions.

Graph

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Solving Quadratic Equations by Graphing

Word Problems

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More Word Problems Using Quadratic Equations - Example 1