a + 1/a = 6 (a≠0) ··· Eq.1
Multiply both sides of Eq.1 by a, then put the equation into the standard form of quadratic. We have:
a²-6a + 1 = 0 Solve the equqtion for a, using the quadratic formula. We have:
a=(6 ± √(6²-4·1·1)) / 2·1 = 3 ± 2√2 We have: a=3+2√2, or a=3-2√2
From Eq.1, 1/a=6-a
A. If a=3+2√2, 1/a=6-(3+2√2)=3-2√2, so a-1/a=(3+2√2)-(3-2√2)=4√2
B. If a=3-2√2, 1/a=6-(3-2√2)=3+2√2, so a-1/a=(3-2√2)-(3+2√2)=-4√2
From A. and B., We have: a-1/a=±4√2
The answer is: If a=3+2√2, a-1/a=4√2, or if a=3-2√2 , a-1/a=-4√2