Trial and improvement

If you are asked to give the solution to an equation to a given number of decimal places (d.p.) or significant figures (s.f.), you can be sure there is no exact solution. In this case, you might be asked to solve it through a method of trial and improvement. The question should indicate the degree of accuracy required (1 d.p., 2 s.f. etc).

Have a look at the example below:

Example

An equation such as x³ + x = 50 does not have an exact solution: the answer is a decimal number. Find the answer correct to 1 decimal place.

Solution

We are looking for a number which, when you cube it and add the number itself, you get the answer 50. One way to do this is to use trial and improvement. Start with a guess, then keep on guessing, trying to get closer to the right answer.

Set it out like this:

• First guess: x = 3
• 3 × 3 × 3 + 3 = 27 + 3 = 30 - too small

• Second guess x = 4
• 4 × 4 × 4 + 4 = 64 + 4 = 68 - too big

• We now know that the answer lies between 3 and 4.
• Third guess: x = 3.5
• 3.5 × 3.5 × 3.5 + 3.5 = 42.875 + 3.5 = 46.375 - too small

• We now know that the answer lies between 3.5 and 4.
• Fourth guess x = 3.6
• 3.6 × 3.6 × 3.6 + 3.6 = 46.656 + 3.6 = 50.256 - too big

We now know that the answer lies between 3.5 and 3.6. But it must be closer to 3.6, so the answer is x = 3.6 correct to 1 decimal place.

Have a go at this one. You will need to have a calculator handy.