There are two pairs (x,y) of real numbers that satisfy the equation x + y = 3, xy = 1. Given that the solutions x are in the form x = (a pm b*sqrt(c))/d where a, b, c, and d are positive integers and the expression is completely simplified, what is the value of a + b + c+ d ?
+129850 x + y = 3
xy = 1 ⇒ y = 1/x
So
x + 1/x = 3 multiply through by x
x^2 + 1 = 3x
x^2 - 3x = -1 complete the square on x
x^2 - 3x + 9/4 = -1 + 9/4
(x - 3/2)^2 = 5/4 take both roots
x - 3/2 = -sqrt (5/4) or x - 3/2 = sqrt (5/4)
x - 3/2 = -sqrt (5) / 2 or x - 3/2 = sqrt (5) /2
x = (3 -sqrt (5)) / 2 x = (3 + sqrt (5) ) / 2
a = 3 b = 1 c = 5 d = 2
Sum = 11
