| Graeme's Number |
This problem was submitted by Graeme MacRae.
Express the following number in simplest form. Please provide explanation with your answer!
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Problem Moderated by: Douglas |
| Problem Solution |
Let X be the first half of Graeme's number, and Y the second half, so
G = X - Y
From this, X3 - Y3 = 20, and XY = 2 (proof left to reader)
Now, cubing the equation G = X - Y gives us:
G3 = X3 - 3X2Y + 3XY2 - Y3
G3 = (X3 - Y3) - 3XY(X - Y)
G3 = 20 - 3(2)G
G3 + 6G - 20 = 0
This third degree equation has one real solution: 2 |
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