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!

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

Example: Cooking

For general cooking you can change things a bit and get away with it, just not too much salt please! Adding an extra 50g of carrot is fine.


Example: Baking

When baking a cake, slight variations in ingredients often won't ruin a recipe, but don't go too far or the result may be a failure.

Adding 100ml extra milk may make the result wet and yucky.

Example: Aerospace Engineering

Calculations for the trajectory and fuel needs of spacecraft require extreme accuracy. Small errors can lead to going off-course and mission failure.

Solving Equations

We still want to be as accurate as possible, so try to handle any approximate data sensibly.

When solving equations:

Like this:

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

Start with:x/7 − 6.3068 + 2π = 0
Subtract −6.3068+2π from both sides:x/7 = +6.3068 − 2π
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 − 6.3068 + 2π = 0

Substitute 0.165 for x:0.165/7 − 6.3068+ 2π = 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


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