- #1
chingel
- 307
- 23
Homework Statement
I am trying to find when the square root of the expression [itex]25+8a^2[/itex] is rational, where the number a also needs to be rational. [itex]\sqrt{25+8a^2}=b[/itex], where a and b are both rational numbers. I am trying to get an expression for a in terms of some other number m, which would always make the number a satisfy the original requirement.
The Attempt at a Solution
I haven't really done a problem like this before, but it looks a lot like Pythagorean triples [itex]5^2+(2\sqrt{2}a)^2=c^2[/itex], so I tried working something with the formula [itex](m^2-n^2)^2+(2mn)^2=(m^2+n^2)[/itex]. I didn't have any other ideas than simply substituting something in there, for example [itex]2\sqrt{2}a=2mn[/itex]; then [itex]m=\sqrt{2}a/n[/itex]; then [itex]2a^2/n^2-n^2=5[/itex], but this just gets me back to the beginning, if I multiply by n^2 and try to solve using the quadratic formula.
Any hints are appreciated. Some examples of a's that work are 0, 10/7 and 15/17.