+62 \(If $a$ and $b$ are nonzero unequal real numbers and $\frac{a-b}{a}=\frac{b}{a-b}$, what is the sum of all possible values for $\frac{a}{b}$?\)
+23246 If (a - b) / a = b / (a - b) then what are the values of a/b?
(a - b) / a = b / (a - b)
cross-multiply: (a - b)2 = a·b
a2 - 2ab + b2 = ab
a2 - 3ab + b2 = 0
Using the quadratic formula (solving for a, using b as a constant):
a = [ - -3b +/- sqrt( (-3b)2 - 4·1·b2 ) ] / ( 2·1 )
a = [ 3b +/- sqrt( 9b2 - 4b2 ) / 2
a = [ 3b +/- sqrt( 5b2 ) / / 2
a = [ 3b +/- sqrt(5)·b ] / 2
a = [3 + sqrt(5)] / 2 · b ---> a/b = [3 + sqrt(5)] / 2
a = [3 - sqrt(5)] / 2 · b ---> a/b = [3 - sqrt(5)] / 2
