Melody

What is  (sqrt3 + sqrt2)^3?

What is  (sqrt3 - sqrt2)^3? 

I did it a super long way, which may or may not be the only way to do it.  

What have you done towards solving it ?

Well, why did not you tell us what you already knew?

Draw a line and put the points on it.   The line can be horizontal and the points do not have to be to scale.

Below are ratios, not actual distances

AP=3    BP=1   where B is between A and P

So

AB=2  and BP=1

Look at the x values,    0,5,10,15

and the y values           5,9,13,17

P(15,17)

Well then rearrange it yourself!

\(x^2+y^2+2xy=30+2xy\\ (x+y)^2=30+2xy\\ \)

You can finish it.

Hint:  The sum of any two sides of a triangle MUST be longer than the 3rd side.

If you take the extreme case where BC and BA are the same tangent.

then angle ABO is 90 degrees, angle BAO=25 degrees  so   angle x must be  180-25-90 = 65 degrees.

There is obviously some theory that Jugoslav knows that says this will always be the case.

I don't know how to prove the general case.

Lets look at a simpler problem

A company produces 10 bags in 5 days.   How much will it poduce in 9 days.

First find out how much it will produce in just 1 day.   How would you do that?

Hang on a tick, is it possible that  \((\sqrt2, \frac{3\pi}{4})\)   is an absolute value and an angle reference?

If so then the rectangular coordinates would be  (-1,1) .  That looks a lot easier to deal with!

\(\frac{a+5i}{1-2bi}=-1+i\\ \)

I solved this and got a=3 and b=2

so     \(a^2+b^3=9+8=17\)