Question

If a-b=5,show that a³-b³-15ab=125.

Answer 1

Answer:

I think given equation means a^3 + b^3 + 15ab = 125 .

It becomes a^3 + b^3 + 15ab - 125 = 0 .

I think you know this identity :

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 .)