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Find all solutions to the system
a + b = 14
a^3 + b^3 = 812.

 

Please help! I don't understand how to do this.

 Nov 16, 2020
 #1
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a + b   =14       (1)

 

Square both sides of  this

 

a^2 + 2ab + b^2 =  196       rearrange  as

a^2 + b^2  =196 - 2ab      (2)

 

a^3 + b^3  = 812

Using the formula  for  the  sum of  cubes we  have that

 

(a + b)  ( a^2 - ab + b^2)  = 812

 

sub in (1)   and (2)

 

(14) ( 196 - 2ab - ab)  = 812         divide both sides by   14

 

(196  - 3ab)  =  58     rearrange as

 

3ab  = 196 - 58

 

3ab  = 138      divide both sides by 3

 

ab = 46

 

b  =   46 / a

 

sub this into 1

 

a + 46/a  = 14     multiply through by a

 

a^2 + 46  = 14a      rearrange as

 

a^2  - 14a  + 46   =   0     complete the square

 

a^2 - 14a  + 49  =   -46 + 49

 

(a - 7)^2  = 3       take  both roots

 

a - 7 =    ±sqrt (3)

 

a =  7 + sqrt (3)        or      a  =  7 - sqrt (3)

 

And  b = the conjugate of either answer

 

So   ( a, b)  =   ( 7 + sqrt (3), 7 - sqrt (3) )  or    ( 7 - sqrt (3) , 7 + sqrt (3))

 

 

cool cool cool

 Nov 16, 2020

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