Suppose $a$ and $x$ satisfy $x^2 + \left(a-\frac{1}{a}\right)x - 1 = 0$. Solve for $x$ in terms of $a$.
+128408 \($x^2 + \left(a-\frac{1}{a}\right)x - 1 = 0$\)
x = [ - (a - 1/a ) ± √ [ ( a - 1/a)^2 - 4(1)(-1) ] ] / 2 simplify
x = [ (1/a -a) ± √ [ a^2 - 2 + (1/a)^2 + 4] ] / 2
x = [ (1/a - a ] ± √ [ a^2 + 2 + 1/a^2 ] ] / 2
x = [ (1/a - a) ± √ [ ( a + 1/a)^2 ] ] / 2
x = [ (1/a - a) ± (a + 1/a) ] / 2
So either
x = [ ( 1/a - a) + (a + 1/a) ] / 2 ⇒ (2/a)/2 = 1/a
Or
x = [ (1/a - a) - ( a + 1/a) ] / 2 ⇒ ( -2a) / 2 = - a
