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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

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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 x^3 + 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:

3 \times 3 \times 3 + 3 = 27 + 3 = 30

  • First guess:
  • x = 3 - too small
  • Second guess x = 4
  • 4 \times 4 \times 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 \times 3.5 \times 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 \times 3.6 \times 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.

Question

Solve the equation y^2 + 2y = 40, correct to 1 decimal place.

Answer

Here is a worked solution.

y^2 + 2y = 40

Let's start with y = 5

5 \times 5 + 2 \times 5 = 25 + 10 = 35 - too small!

 

y = 6 ?

6 \times 6 + 2 \times 6 = 36 + 12 = 48 - too big!

So the answer lies between 5 and 6.

y = 5.5 ?

5.5 \times 5.5 + 2 \times 5.5 = 30.25 + 11 = 41.25 - too big!

 

y = 5.4 ?

5.4 \times 5.4 + 2 \times 5.4 = 29.16 + 10.8 = 39.96 - too small!

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

y = 5.4 (1 d.p.)

So y = 5.4

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