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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.

 Feb 26, 2022
 #1
avatar+23198 
+2

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.

 Feb 26, 2022
 #2
avatar+36433 
0

A different route:

 

(a+b) = ( 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    

 Feb 26, 2022

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