Representation of mathematically “Completing the square”.

Complicated or not, visual demonstrations are great and the colours are incredibly appealing



Representation of mathematically “Completing the square”.

Complicated or not, visual demonstrations are great and the colours are incredibly appealing


Fibonacci you crazy bastard….

As seen in the solar system (by no ridiculous coincidence), Earth orbits the Sun 8 times in the same period that Venus orbits the Sun 13 times! Drawing a line between Earth & Venus every week results in a spectacular FIVE side symmetry!!

Lets bring up those Fibonacci numbers again: 1, 1, 2, 3, 5, 8, 13, 21, 34..

So if we imagine planets with Fibonacci orbits, do they create Fibonacci symmetries?!

You bet!! Depicted here is a:

  • 2 sided symmetry (5 orbits x 3 orbits)
  • 3 sided symmetry (8 orbits x 5 orbits)
  • sided symmetry (13 orbits x 8 orbits) - like Earth & Venus
  • sided symmetry (21 orbits x 13 orbits)

I wonder if relationships like this exist somewhere in the universe….

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Some formulas are plainly beautiful, aren’t they?

Suppose our angle is in the interval (90*,180*). Is the cos[(angle)/2] positive or negative?

If our angle is somewhere between 90° and 180°, that means that angle/2 is somewhere between 45° and 90°

Looking at the graph of the cosine function helps with this:

45° translates into π/4 (radians)
90° translates into π/2
180° translates into π

In the following picture the interval for our angel is orange, the interval for angle/2 blue:

As you can see the cosine-value for the blue interval is always positive, so there you go!

- Frauke

how to form quadratic equations in standard form?

The standard form for quadratic equations looks like this:

ax²+ bx + c = 0

To get there, you will usually have to expand (undo brackets) and move everything to one side so you have 0 on one side.

To find some examples, check here or here.

Hope this helps!


The lengths of two sides of an isosceles triangle are 8 and 10. The length of the third side could be: a) 6, only .. b) 8, only .. c) 10, only .. d) 8 or 10

Since an isosceles triangle is by definition “triangle with at least two equal sides”, your answer is d) 8 or 10, because they could be 8,8,10 or 8,10,10, and it would in any case be still isosceles. Since we don’t have any indication of the equal sides being the longer ones or the shorter ones, we can’t say for sure whether they need to be 8 or 10 :)



How to read math. You’d be surprised how far this will get you.

EDIT: Some corrections

David stands near a tree that casts a 35-ft. shadow, while David’s shadow is 10 feet in length. If David is 6 feet tall, how tall is the tree?

All you need is to do a proportion (you can also look at the question I just answered) 

So you now that a 6ft tall person casts a 10ft tall shadow, and a x tall tree casts a 35ft shadow

You can also write it as 6:10=x:35 -> (6*35):10 -> the tree has to be 21ft to fit your problem :)


The scale on a map indicates that 1 2 3 inches represent 35 actual miles. i) How many miles does eight inches represent? ii) If the distance between two towns is 75 miles, how far apart are they on the map

Assuming that 3 inches are 35 actual miles (If I am assuming wrong please correct me, but I don’t really understand what you meant by “1 2 3 inches”), and you need to find out how much 8 inches would be, you just need to put down a proportion.

So you have that 3in:35m=8in:x

So you have (35*8):3= 93,(3)m, which fits, if you consider that 3in are 35m, so 6 inches would be 70m and 8 a bit less than 105m!

Then you need to know how much 75miles would be, and you do just the same thing. You put down the proportion this way


So you have (3*75):35= 6,42~inches

Hope I helped!


Is 2 a solution of 3x+5 ≥ 12?


3(2) + 5  ≥ 12 ?
6 + 5  ≥ 12 ?
11  ≥ 12 ?

No, 11 is not greater or equal than 12.