# Standard Form

### Unit 3

## Summary

## Completing the Square:

(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:

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.

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

## 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

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