Start by writing equations! Since the problem asks "what is the length of the edge", write "Let x be the length of a side of the cube, in centimeters".

Then the volume of that cube is, of course,

.

Increasing one side by 6 cm changes from x to x+ 6.

Increasing a second side by 12 cm changes from x to x+ 12.

Decreasing the third side by 4 cm changes from x to x- 4.

The volume of this rectangular solid is (x+6)(x+ 12)(x- 4) and we are told that is double the original volume:

and multiplying that by x- 4 gives

So we have

which is the same as

. We need to solve that equation.

Now, cubics, in general, are

**hard** but we notice that the coefficient of

is 1 so any

**rational** number solution, if there is one, must be an integer factor of 288. Crossing our fingers and hoping there is such a solution, we try x= 1, 2, 3, 4, 6, etc. and fortunately find that

. x= 6 is a solution. From that it is not to difficult to find that

and that quadratic has no real number solutions. The only solution is x= 6.

Now be sure to write the answer clearly:

The cube must have edges of length 6 cm.

and check:

If the original cube has edge length 6 cm. then its volume is

= 216 cubic centimeters.

Increasing the one length by 6 cm, another by 12 cm, and reducing the third by 4 cm would give a rectangular solid (6+6) by (6+ 12) by (6- 2)= 12 by 18 by 2 and that has volume (12)(18)(2)= 432 cubic centimeters = 2(216) cubic centimeters, as we said.