Question

If 5+2√3 upon 7+√3=a-√3b , find a and b where a and b are rational numbers.

Answer 1

LHS:
(5+2√3)/(7+√3)
=(5+2√3)(7-√3) / (7+√3)(7-√3)
=(35 + 14√3 - 5√3 -6) / (49-3)
=(29+9√3) / (46)
=(29/46) + (9√3/46)
=(29/46) - (-9√3/46)

So on comparing both sides
a=29/46
and b=-9/46

Answer 2

Given
(5 + 2√3) / (7 + √3) = a - b√3

By rationalization, we get

(5 + 2√3)(7 +√3) / (7 +4 √3) (7 - √3) = a - b√3

(35 - 5√3 + 14√3 – 6) /49-3= a - b√3

(29 + 9√3 ) /46= a -b√3

On equating the above equation, we get

a = 29/46 and b = -9/46