If *a* ≠ *b*, *a*^{3} – *b*^{3} = 19*x*^{3}, and *a* – *b* = *x*, which of the following conclusions is correct?

*a*= 3*x**a*= 3*x*or*a*= -2*x**a*= -3*x*or*a*= 2*x**a*= 3*x*or*a*= 2*x**a*= 2*x*

Given *a* ≠ *b*, *a*^{3} – *b*^{3} = 19*x*^{3}, and *a* – *b* = *x*. We need to find *a* in terms of *x*.

This means *b* will have to go, so we note now that *a* – *b* = *x* → *b* = *a* – *x*.

Now we take a deep breath.

a^{3} – b^{3} = |
(a – b)(a^{2} + ab + b^{2}) |

19x^{3} = |
x (a^{2} + a (a – x) + (a – x)^{2}) |

19x^{3} = |
x (a^{2} + a^{2} – ax + a^{2} – 2ax +x^{2}) |

19x^{3} = |
x (3a^{2} – 3ax + x^{2}) |

19x^{3} = |
3a^{2}x – 3ax^{2} + x^{3} |

0 = | 3a^{2}x – 3ax^{2} – 18x^{3} |

0 = | 3x (a^{2} – ax – 6x^{2}) |

0 = | 3x (a – 3x)(a + 2x) |

So *a* = 3*x* or *a* = -2x, which is candidate (b). Whew.