Find the largest possible value of x in the simplified form x=\frac{a+b\sqrt{c}}{d} if 5x/6 +1= 3/x - 1, where a,b,c,$ and d are integers. What is acd/b?
+129907 \( x=\frac{a+b\sqrt{c}}{d} if 5x/6 +1= 3/x - 1\)
5x / 6 + 1 = 3/x -1 multiply through by the common denominator of 6x
5x^2 + 6x = 18 - 6x rearrange as
5x^2 +12x - 18 = 0
Using the Q Formula we have x =
-12 + sqrt [ 12^2 - 4 (5)(-18) ] -12 + sqrt [ 144 + 360] -12 + sqrt [ 504]
_________________________= ____________________ = ________________ =
2 (5) 10 10
-12 + 6 sqrt [ 14 ] -6 + 3sqrt [14 ]
_________________ = __________________
10 5
acd / b = (-6) (14) (5) / 3 = -140
