tackle this and read v as root over.?

Question

(v3) ^ (2) = (14 divided by 3) ^ (2) - (13 divided by 3) ^ (2)
if (v2) ^ (2) = c ^ (2) - b ^ (2)
find c and b (c and b must be a perfect square and a rational number.)
if possible find for (v5) ^ (2) also .

Answer 1

c = 3/2, b = 1/2.
Basically, you need to find
2 = c^2 - b^2 in the rationals. But, since finding rational solutions is annoying, it's easier finding integer solutions to the equation
2a^2 = c^2 - b^2
2a^2 = (c-b)(c+b)
Since c+b and c-b have the same parity, they need to both be even. This means that one of them is divisble by 4. So we need to have a = 2k for some k.
So we actually have to solve 8k^2 = (c-b)(c+b), but 8 factors as 4 * 2, so let c+b = 4, c-b = 2, which gives c = 3, b = 1. But you actually want c/a, b/a, and a = 2, so c= 3/2, b = 1/2 works.
For 5, we can just take c = 3, b = 2, and we have that 5 = 9 - 4.

Answer 2

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