If a ≠ b, a3 – b3 = 19x3, and a – b = x, which of the following conclusions is correct?
- a = 3x
- a = 3x or a = -2x
- a = -3x or a = 2x
- a = 3x or a = 2x
- a = 2x
Given a ≠ b, a3 – b3 = 19x3, 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.
| a3 – b3 = | (a – b)(a2 + ab + b2) |
| 19x3 = | x (a2 + a (a – x) + (a – x)2) |
| 19x3 = | x (a2 + a2 – ax + a2 – 2ax +x2) |
| 19x3 = | x (3a2 – 3ax + x2) |
| 19x3 = | 3a2x – 3ax2 + x3 |
| 0 = | 3a2x – 3ax2 – 18x3 |
| 0 = | 3x (a2 – ax – 6x2) |
| 0 = | 3x (a – 3x)(a + 2x) |
So a = 3x or a = -2x, which is candidate (b). Whew.