A funny algebra puzzle!

a=3 and b=0. a.b = 3.0 (the . indicates the decimal and not a multiplication!). 3² - 0² = 9

Please note the dot in the relation a.b is well a multiplication! It is not the decimal sign... Sorry to have not been highlighting this...

Sorry Fons, the dot used here is well a multiplication sign... Sorry for not having highlighting this...

a=3, b=0

Ah, ok. I made the same "assumption", thanks for claryfing.

a = +- 3.1477

b = +- 0.953

a²-b² = 9 => b² = a²-9
a.b = 3 => b = 3/a => b² = 9/a²
a²-9=9/a²
a4-9a²-9 = 0
(a²-4.5)²-20.25-9=0
a²-4.5 = SQRT(29.25)
a = SQRT(SQRT(29.25)+4.5) = 3.1477, b = 3/a = 0.9530

By substitution, you get y*y-9*y-9=0, with y=a*a. By solving, you get y1=9.9083 and y2=-0.9083.

y2 is not acceptable, then you get a=+/- 3.1477 and b=3/a=+/-0.953

Here we can form on equation that is (a+b) ^2 - (a-b) ^2 = 4ab, 

And (a+b) = 9/(a-b) so one quadratic will be formed that is (9/(a-b)) ^2 - (a-b) ^2 = 12 . So after solving this we will get the value of a-b as 2.194 and then value of a+b will be 4.102 after precision. 

Then we can find the value of a and b as 3.148 and 0.954.

a=5        b=4

5²=25    4²=16     25-16=9

in a  rectangular triangle with hypotenuse = 5 and opposite side = 4 the adjacent side =3