Algebra, Done!

I just finished all of algebra!

The last thing I did in algebra was conic sections.

In conics we learned how to graph ellipses, hyperbolas, circles, and parabolas.

This a line:

y = x + 5

Parabola:

y = -3x² - 6x - 1

Circle:

(x - 2)² + (y + 3)² = 9

Ellipse:

(x²/6) + (y²/16) = 1

Hyperbola:

x² - y² = 39

A parabola is U-shaped. But it can open up or down.

An ellipse is an oval.

Hyperbola is like two parabolas.

You can tell what shape an equation is just by looking at it.

If one variable either ,x or y, is squared it is a parabola.

If both variables are squared then the graph might be a circle, an ellipse, or a hyperbola.

If the coefficients of x² and y² are the same sign AND equal, the equation represents a circle.

If the coefficients of x² and y² are the same sign BUT different, you have an ellipse.

If the coefficients of x² and y² have different signs(one positive and one negative), then the graph will be a hyperbola.

I did this with my dad:

Now I'm exited to start calculus and advanced geometry.

Exponents

Today I was doing exponents.

These are their properties:

x^a · x^b = x^a+b - Multiplying Powers

(x^a)^b = x^ab - Raising a Power to a Power

x^-n =1/xⁿ - Negative Exponents

Examples:

x³ · x⁷ = x³⁺⁷ = x¹⁰

(x⁵²)² = (x⁵²)² = x¹⁰⁴

x^-4 = 1/x⁴

3² · 3³ = 3⁵ = 243

(2³)² = 2⁶ = 64

4^-2 = 1/4² = 1/16

Imaginary Numbers

Yesterday I learned about i. We use it to describe imaginary numbers.

This is the definition of i:

i = √ -1 


Now if that is true, then the following is true:

i² =i· i = -1

i³ = i² · i = -i

i⁴ = i³ · i = 1

i⁵ = i⁴ · i = i

i⁶ = i⁴ · i² = -1

i⁷ = i⁴ · i³ = i³ = -i

i⁸ = i⁴ · i⁴ = 1

i⁹ = i⁸ · i = i

i¹⁰ = i⁸ · i² = -1

As you can see, every 4th power of i = 1.

So you can just divide i's exponent by 4 and the remainder will tell you the answer {1,i,-1,or -i}.

These are the questions I did yesterday.

A complex number has two parts - a real part and an imaginary part.

For example:

3 + 6i

I'll write another post on them later.

More Algebra

Now I am studying Polynomials.

Here is an example:

6x² - 7x + 1

Here is another example:

10x⁵ + 4x⁴ - 13x³ + .5² - 102

Each part of a Polynomial is called a Monomial. My dad calls them "terms".

The number in front of each term is called a coefficient.

The little numbers above the x's are called exponents.

When x-terms have the same exponent they can be easily combined by adding their coefficients.

For example:

3x⁴ - 17x⁴ = -14x⁴

(-1/3)x⁵ + (1/2)x⁵ = (1/6)x⁵

Two terms or two monomials, make a "binomial". (Bi - means two).

You can also multiply, divide, and factor Polynomials

Linear Equations

In algebra now I'm solving some equations with 2 variables.

For example:

x + y = 10

Here are some of the solutions to this equation:

(1,9)
(-3,13)
(9,1)
(1.5,8.5)
(13,-3)
(1000,-990)
(0,10)
(-33.2,43.2)
.
.
.

As you can see there are infinite solutions (x,y) to this equation. In fact, there are infinite solutions to any equation with an x and y.

The best way to describe these solutions is with a graph.

This graph is always a straight line that extends in both directions to infinity.

The easiest way to draw these lines is by computing the x and y intercepts.

All you have to do is set x equal to 0 and solve for y. Then set y equal to 0 and solve for x.

Then you have two points that you can connect to draw the infinite line!

For example, the intercepts of x + y = 10 are (0,10) and (10,0).

The other common way to graph a line is with slope-intercept form: y = mx + b where m is the slope of the line and b is the y-intercept.

Thanks!

I want to thank everybody who has been ordering through the Amazon ads on my site and my Dad's site.

So far my commission for this month is already around $50.

Since I earn 6% of all orders, that means $50 is 6% of the total.

X = total

.06 · X = 50

Divide both sides by .06

X = 50 / (.06)

X = 833.33

So the total spent through my ads this month is about $833.

Thanks so much!

Finished Another Book!

I just finished this book - Singapore Math 6B.

It covered:

Rate

Circle

Pie Charts

Area and Perimeter

Volume

Challenging Word Problems

Rate = Distance / Time

Circumference = Π * Diameter

Area of a Circle = Π * Radius²

Area of a Semi-Circle = .5 * Π * Radius²

Area of a Quadrant(of a Circle) = .25 * Π * Radius²

Volume of a Box = Width * Length * Height

Volume of a Cube = Side³

How Many Seconds Are In A Century?

Today I asked my dad how many seconds are in a century.

So he took out a piece of paper and we figured it out.

And here is our equation:

60sec/1minute · 60min/1hour · 24hours/1day · 365days/1year · 100years/1century = seconds in a century

This is called the factor-label method. All those fractions after the first one are actually equal to one.

When we multiply it out, most of the labels cancel out:

60sec/1minute · 60min/1hour · 24hours/1day · 365days/1year · 100years/1century = seconds in a century


Now we have to do the arithmetic:

60 · 60 · 24 · 365 · 100 = seconds in a century


We tried doing it on a calculator, but the answer was too big!

So we factored it:

36 · 24 · 365 · 10⁴


And multiplied the first part on the calculator:

315,360 · 10<sup>4

Then we added four zeroes:</sup>


3,153,600,000

There are over 3 billion seconds of  in a one hundred year period!

Rate * Time = Distance

If you go 70mph for one hour....you'll go 70 miles.

If you go 70mph for two hours....you'll go 140 miles.

If you go 70mph for half an hour.... you'll go 35 miles.

The rule is:

rate * time = distance

In my algebra book sometimes they mix it up.

Sometimes they give the distance and the rate and have to solve for time.

Sometimes they give the distance and the time and have to solve for rate.

That is easy:

rate = distance / time

time = distance / rate

Another thing they want me to do is to find the average speed:

average speed = total distance / total time

Math Tricks

Did you that:

a·(b+c) = a·b + a·c ?

It's called the distributive law of equality.

You can use it to make hard math questions easier.

For example:

19·20 = ?

Well, I know that 19·20 = (20-1)·20

And I know that 20·20 = 400

So 19·20 = 20·20 - 1·20 = 380

I say out loud, "nineteen twenties equals twenty twenties minus one twenty - which equals four hundred minus twenty - which equals three hundred and eighty."

I can do that in my head!

Basic Algebra

11X + 23 = 4X - 5

That ↑ is called an equation.

My job is to find out the mystery number X.

First I get all the X's on one side.

So I subtract 4X from both sides of the equation.

11X + 23 - 4X = 4X - 5 - 4X

7X + 23 = -5

Next I subtract 23 from both sides.

7X + 23 - 23 = -5 - 23

7X = -28

Now I need to turn the 7 into an invisible 1.

So we divide both sides by 7.

7X / 7 = -28 / 7

X = -4


Let's check our answer.

11·-4 + 23 = 4·-4 - 5

-44 + 23 = -16 - 5

-21 = -21

My Second Spreadsheet

That is my second spreadsheet ↑!!

It is a geometry calculator that will tell you the area of different shapes.

Click here to see my other calculator.