SOLUTION: If (1 + a)(1 + b)(a + b) = 2022 and a³ + b³ = 1993 Find (a + b)

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Question 1208090: If (1 + a)(1 + b)(a + b) = 2022
and a³ + b³ = 1993
Find (a + b)
Found 2 solutions by EPMcCravy, ikleyn:
Answer by EPMcCravy(4)   (Show Source): You can put this solution on YOUR website!

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
If (1 + a)(1 + b)(a + b) = 2022 and a^3 + b^3 = 1993, find (a + b).
~~~~~~~~~~~~~~~~~~~~~~~~~~~


        Be ready:   you will get a nice solution right now. 


First,  (1 + a)(1 + b)(a + b) = (1 + a + b + ab)*(a+b) = (a+b) + (a+b)^2 + ab*(a+b).

Next,  a^3 + b^3 = (a+b)*(a^2 - ab + b^2) = (a+b)*((a+b)^2-3ab) = (a+b)^3 - 3ab*(a+b).



Thus you have these two equations

    (a+b) + (a+b)^2 + ab*(a+b)  = 2022    (1)

    (a+b)^3         - 3ab*(a+b) = 1993    (2)



Multiply equation (1) by 3; keep equation (2) as is.  Then add them.  You will get

    3(a+b) + 3(a+b)^2 + (a+b)^3 = 3*2022 + 1993,

or

    3(a+b) + 3(a+b)^2 + (a+b)^3 = 8059.



Add 1 to both sides

    1 + 3(a+b) + 3(a+b)^2 + (a+b)^3 = 8060.


Notice that the left side is ((a+b)+1)^3.


So, the last equation is

    ((a+b)+1)^3 = 8060.



Take cube root of both sides

    (a+b)+1 = .



From this, get

     a+b =  -  = 19.04987552  (rounded).



ANSWER.  If (1 + a)(1 + b)(a + b) = 2022 and a^3 + b^3 = 1993,  then  a+b =  -  = 19.04987552  (rounded).

Solved.

Thus you have both the exact formula and an approximate value.



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