Q.176. Algebra, Diophantine equation

Question 176.
Find a + b if a^3 + b^3 = 8 - 6ab.

Answer 176.
a^3 + b^3 = 8 - 6ab
=> (a + b)^3 - 3ab(a + b) = 8 - 6ab
=> (a + b)^3 - (2)^3 = 3ab (a + b - 2)
=> (a + b - 2) [(a + b)^2 + 2(a + b) + 4] - 3ab (a + b - 2) = 0
=> (a + b - 2) [(a + b)^2 + 2(a + b) + 4 - 3ab] = 0
=> a + b - 2 = 0 or (a+b)^2 + 2(a+b) + 4 - 3ab = 0
=> a + b = 2   ...   ( 1 )

OR
(a+b)^2 + 2(a+b) + 4 - 3ab = 0
=> a^2 + (2 - b)a + b^2 + 2b + 4 = 0
=> a = (1/2) [(b - 2) ± √[(2 - b)^2 - 4(b^2 + 2b + 4)]]
=> a = (1/2) [ (b - 2) ± √(- 3(b + 2)^2]
As the discriminant ≤ 0, for real solution it has to be zero
=> b = - 2
=> a = - 2
=> a + b = - 4   ...   ( 2 )

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