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! .
If (1 + a)(1 + b)(a + b) = 2022 and a^3 + b^3 = 1993, find (a + b).
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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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