Trial and Improvement

Trial and Improvement questions will always follow the same pattern - only the values will change.

Question

The solution to   x^3 - x = 94    has a solution between 4 and 5.
Using Trial and Improvement, find the value of x.
Give your answer correct to one decimal place.

As the question says, the solution (the value of x we need) of the equation  x^3 - x = 94  is between 4 and 5. Trial and Improvement means you must try different values of x until you find one that works for the equation. Don't forget, this question will be on your calculator paper, so remember to use it!

Start by checking the values x=4 and x=5:

x = 4
x^3 - x = 4^3 - 4 = 60

We want x^3 - x = 94, so '60' is too small.

x = 5
x^3 - x = 5^3 - 5 = 120  (too big)

So what value should we try next? x = 4.5 would be good, as it is half-way between 4 and 5.

x = 4.5
x^3 - x = 4.5^3 - 4.5 = 86.625  (too big)

In the exam, write the values you have tried in a table, like this:

x        x^3 - x

4         60          Too small
5         120        Too big
4.5      86.625   Too small

If you continue to try values in this way, you will find that the solution lies between x = 4.6 and x = 4.7.
It is important to remember that you must "Give your answer correct to one decimal place", as the question states. This means the solution will be either 4.6 or 4.7 (because any number between 4.6 and 4.7 will round to either 4.6 or 4.7, accurate to one decimal place).

It makes sense to check x = 4.65 (half-way between 4.6 and 4.7)

x = 4.65
x^3 - x = 4.65^3 - 4.65 = 95.894625  (too big)

This means the solution to the equation is between 4.6 and 4.65.

Any value between 4.6 and 4.65 will, when rounded to 1 decimal place, equal 4.6.
E.g. 4.62 = 4.6 (1 decimal place), 4.649999999999 = 4.6 (1 decimal place)
Therefore, our solution, accurate to one decimal place, is x = 4.6 (because the solution is closer to 4.6 than 4.7).

Remember: You will normally have to give your answer accurate to one decimal place, or 3 significant figures, etc. Make sure you do this at the end!

Top Tip: Don't forget to show all the values you have tried. Even if you do not get the final answer correct, you can still get most of the marks if you have shown all the values of x you have checked.

MH