Question

If a+b=5 then a^3+b^3+15ab=?

Answer 1

x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz) 
Let x = a , y = b and z = -5 . So 
a^3 + b^3 - 125 + 15ab = (a + b - 5)(a^2 + b^2 + 25 - ab + 5a + 5b) 

Therefore , given equation becomes 
(a + b - 5)(a^2 + b^2 + 25 - ab + 5a + 5b) = 0 

Case1. a + b - 5 = 0 
This means a + b = 5 . 

Case2. a^2 + b^2 + 25 - ab + 5a + 5b = 0 
We can think 
(a + 5)^2 + (b + 5)^2 - (a + 5)(b + 5) 
= a^2 + 10a + 25 + b^2 + 10b + 25 - ab - 5a - 5b - 25 
= a^2 + b^2 - ab + 5a + 5b + 25 
= 0 
So let A = a + 5 and B = b + 5 , 
A^2 + B^2 - AB = 0 
It is a quadratic equation of B , so 
B = (1/2)(A ± √(A^2 - 4A^2)) 
= (1/2)(A ± √(-3A^2)) 
A and B must be real , so the solution is A = 0 and B = 0 only . 
This means a = -5 and b = -5 , a + b = -10 . 

So , a + b = 5 or -10 . 
(a + b = -10 only when a = -5 and b = -5 .)

Answer 2

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