If 8a³ + b³ + 27c³ = 18 abc, then what is the relation among a, b and c?

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Accepted Answer

Given the equation: 8a³ + b³ + 27c³ = 18abc

We recognize this as a form of the sum of cubes identity:

x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

If the expression equals 3xyz, then the factorization simplifies when x + y + z = 0.

Rewrite the given equation as:
8a³ + b³ + 27c³ - 18abc = 0

Let x = 2a, y = b, z = 3c

Then:
x³ + y³ + z³ - 3xyz = 0

This implies:
x + y + z = 0

Substitute back:
2a + b + 3c = 0

Rearranged:
b + 3c = -2a

However, this matches option 4. But the original equation is:
8a³ + b³ + 27c³ = 18abc

Since the identity is x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx), and for the expression to be equal to 3xyz, the factor (x + y + z) must be zero.

Therefore, the relation is:
2a + b + 3c = 0
or
b + 3c = -2a

Hence, option 4 is correct.

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