Simple Cubic Equations

Cubing a number means multiplying it by itself three times. Four cubed is four times four times four, which is sixty-four. The cube root undoes that: the cube root of sixty-four is four. Negative numbers work too. Negative three cubed is negative three times negative three times negative three. The first two make positive nine, and nine times negative three is negative twenty-seven. So the cube root of negative twenty-seven is negative three. Unlike square roots, cube roots of negative numbers are fine.

Compare a square with a cube. x squared equals four has two solutions, two and negative two, because both of them square to four. You can see it on the graph: the parabola meets the line y equals four twice. But x cubed equals negative eight has only one solution, negative two, because two cubed is positive eight, not negative eight. The graph of y equals x cubed keeps climbing, so it meets any flat line exactly once. So x cubed equals two hundred and sixteen has just one solution, x equals six.

Before you take a cube root, get x cubed on its own. In two x cubed equals two hundred and fifty, the two multiplies x cubed, so divide both sides by two: x cubed equals one hundred and twenty-five, and x is five. In negative three x cubed equals twenty-four, divide by negative three: x cubed equals negative eight, so x is negative two. Check each one by putting it back in. Remember that two x cubed means two times x cubed. Only x is cubed, not the two.

When the cube is on a whole bracket, treat the bracket as a single block. For x minus one, all cubed, equals twenty-seven, take the cube root of both sides: x minus one equals three, so x is four. For x plus four, all cubed, equals negative sixty-four, the cube root is negative four, so x plus four equals negative four and x is negative eight. If there is a number in front, divide it off first: two times x plus three cubed equals sixteen gives x plus three cubed equals eight, so x plus three is two and x is negative one.

Cubic equations come up whenever you know the volume of a cube. The volume is the side length cubed, so the side is the cube root of the volume. A cube with volume three hundred and forty-three cubic centimetres has a side of seven centimetres, because seven cubed is three hundred and forty-three. If a cube has side x plus two and volume five hundred and twelve, then x plus two is the cube root of five hundred and twelve, which is eight, so x is six. Quiz and exercises now.