Approximate Solutions

Sometimes it is difficult to solve an equation exactly. But an approximate answer may be good enough!

What is Good Enough?

Well, that depends what you are working on!

  • If you are dealing with millions of dollars then you should try to get pretty close indeed. that might need many significant digits.
  • If you are calculating how much food to buy for a party, then a small error won't matter so much. You could always buy extra to be sure.
  • Or something in between!

So understanding what you are working on helps you know how accurate you should be.

Solving Equations

To help reduce error, when solving equations:

  • first solve for x = something
  • then do any calculations

Like this:

Example: Solve x/7 + 2π - 6.3068 = 0   (to 3 decimal places)

Start with:   x/7 + 2π - 6.3068 = 0
Move 2π - 6.3068 to the right:   x/7 = - (2π - 6.3068)
Multiply by 7:   x = 7(6.3068 - 2π)
NOW, do the calculations:   x = 0.165

Why wait until the end to do the calculations? Well, every time you do a calculation you can introduce an error. If you do this several times your errors can accumulate to be quite large.


If your answer is approximate, then your checking will also be approximate

Example: Check that x = 0.165 solves x/7 + 2π - 6.3068 = 0

Substitute 0.165 for x:   0.165/7 + 2π - 6.3068 = 0
Calculate:   -0.00004 = 0

Not quite right, but very close.

Graphical Estimation

You can make good approximations using graphs, particularly by using a zoom function, like on our Function Grapher.

Here is an example:

Example: estimate the solution to x3 - 2x2 - 1 = 0 (to 2 decimal places).

Solution: Plot it!

Here is my first attempt. I can see it crosses through y=0 at about x=2.2


Let us zoom in there to see if we can see the crossing point better:


It crosses between 2.20 and 2.21 ... slightly closer to 2.21. We are asked for 2 decimal places, so our answer is:


x3 - 2x2 - 1 = 0 at about x=2.21


Check: (2.21)3 - 2(2.21)2 + 2 = approx 0.025, close to y=0