If a + b = 7 and a^3 + b^3 = 44, what is the value of the sum 1/a + 1/b? Express your answer as a common fraction.
+23252 This may get you started:
a + b = 7 ---> b = 7 - a
a3 + b3 = a3 + (7 - a)3 = 44
Expanding (7 - a)3 = (7)3 - (3 72·a) + (3·7·a2) - a3 = 343 - 147a + 21a2 - a3
Substituting:
a3 + b3 = a3 + 343 - 147a + 21a2 - a3 = 44
343 - 147a + 21a2 = 44
21a2 - 147a + 343 = 44
21a2 - 147a + 299 = 0
Use the quadratic equation to find a; use that answer to find b; and finish.
+37153 A different route:
(a+b)3 = ( a^3 + b^3 ) + 3 a^2b + 3 b^2 a
73 = ( 44 ) + 3 ab ( a+b) we know (a+b) = 7 .... sub that in to get:
7^3 - 44 = 21 ab
ab = 299/21
Now 1/a + 1/b = (a+b) / ab
= (7) / (299/21) = 147/299
