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Home > Maths II > Relationships > Trial and improvement

Maths II

Trial and improvement

If you are asked to solve an equation where there is no exact answer, you might need to use trial and improvement.

# 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 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:
• - too small
• Second guess
• - too big
• We now know that the answer lies between 3 and 4.
• Third guess:
• - too small
• We now know that the answer lies between and .
• Fourth guess
• - too big

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

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

Question

Solve the equation , correct to 1 decimal place.

Here is a worked solution.

- too small!

- too big!

So the answer lies between 5 and 6.

- too big!

- too small!

So the answer lies between 5.4 and 5.5, but must be closer to 5.4

(1 d.p.)

So

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