Question

If a + b + 2c = 0, then prove that a³ + b³ + 8c³ = 6abc.

Answer 1

Given: a + b + 2c = 0

To Prove: a³ + b³ + 8c³ = 6abc

Proof:

1. From the given, define a new variable:

x = a, y = b, z = 2c

2. Using these, the given equation becomes:

x + y + z = 0

3. Recall the standard identity for three variables whose sum is zero:

x³ + y³ + z³ = 3xyz if x + y + z = 0

4. Substitute back:

a³ + b³ + (2c)³ = 3 × a × b × (2c) 

a³ + b³ + 8c³ = 6abc