If (3 + √5)/(3 − √5) = a + b√5 with a and b rational, then b equals:

Multiply numerator and denominator by the conjugate 3 + √5: the denominator becomes 9 − 5 = 4 and the numerator becomes $(3+\surd 5{)}^2=14+6$√5. So the value is (14 + 6√5)/$4=7/2+(3/2)$√5, giving $b=3/2$. (7/2 is a rather than $b$; 6 forgets to divide by 4 at all; 3 divides the 6 by 2 instead of by 4.) Numerically $(3+2.236)/(3-2.236)$ ≈ 6.854, and 3.5 + 1.5(2.236) ≈ 6.854.