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Thinking-Out-Of-The-Box Mathematics

The dimensions of a rectangular box also happen to be the roots of the cubic equation x3 + 6x2 +9x - 2 = 0.

Find the total surface area of the rectangular box....without finding the roots of the equation!


Source: Charlotte Hartman...a great problem solver!

Hint: Consider the general quadratic: x2 + px + q = 0, with roots -r and -s.

What do you know: rs = q and r + s = p.

Now, what about the general cubic.....


Solution Commentary: Suppose -r, -s, -t are the roots of the general cubic roots of x3 + px2 +qx + z = 0. Then, we know z = rst, q = rs + st + rt, and p = r + s + t. This can be shown by multiplying out the factors (x + r)(x + s)(x + t).

Do you start to see a pattern here...what would you expect to be true for the general quartic?

Back to the original problem at hand, if r, s, t are the roots of x3 + 6x2 +9x - 2 = 0, then rs + st + rt = 9. But, the dimensions of the rectangular box then are r, s, t as well...and its total surface area is 2(rs + st + rt) = 2(9) = 18.

Math can be powerful!