In the examples above, you have seen that cubic equations can have 3 real roots but this is not always the case and as you see below, a cubic equation can in fact have 1, 2 or 3 roots.
The other problem encountered above is knowing the value of one of the roots. There are methods that we can use to determine all the roots but this is rather more complex than the procedure for finding the roots of a quadratic equation.
The following graphs show how it is possible to have 3, 2 or 1 root as the roots are given by the x-value when the graph crosses the axis (and the value of the cubic is zero).
Note also that when the coefficient of the x3 term is negative, then the curve looks like the one opposite; as before there will be
3 (as in the illustration), 2 or 1 root.
A practical way of finding the 3 roots is to consider a numerical or graphical approach.
For example, to solve the equation
x3 − 100x2 + 2000x − 1500 = 0
it is a help to work out a few values of the function first. Let the function be labelled f(x).
A table of values is shown on the right.
The arrows show where
f(x) changes from positive to negative or vice versa.
Hence the zeros of f(x) must occur
between 0 and 10
between 20 and 30
between 70 and 80
Trial and improvement, or graphs in the correct regions, give the answers of 0.78, 26.40 and 72.82, working to 2 decimal places.
Worked Examples
Exercises
