Solving a cubic equation using trial and improvement method (to 1 decimal place)

In this article we shall be solving a cubic equation using trial and improvement. Trial and improvement is like a guessing method of solving equations – pick a number and improve your answer.

Example 1

Use trial and improvement to solve this cubic equation to 1 decimal place:

x³ -2x  = 30

The equation has a solution between x = 3 and x = 4.

First draw up a table. Although you are told that x is between 3 and 4 you must substitute 3 and 4 into your equation to guarantee full marks. Also note that 30 is your target number.

x , x³ -2x  = 30 , comment

3 , 3³ - 2×3 = 21 , too small

4 , 4³ - 2×4 = 56 , too big

Now go half way between x =3 and x =4. So try x = 3.5.

x , x³ -2x  = 30 , comment

3.5 , 3.5³ - 2×3.5 = 35.875 , too big

This is too big so try x = 3.4.

x , x³ -2x  = 30 , comment

3.4 , 3.4³ - 2×3.4 = 32.504 , too big

Again x =3.4 is too big so let’s try something smaller.

x , x³ -2x  = 30 , comment

3.3 , 3.3³ - 2×3.3 = 29.337 , to small

Now the final answer will either be x = 3.4 or x =3.3. To confirm which is the closest solution go half way between 3.3 and 3.4. You must do this to guarantee full marks in your exam.

x , x³ -2x  = 30

3.35 , 3.35³ - 2×3.35 = 30.895375

Since 30 is between 29.337 and 30.895375, then the final answer is x = 3.3.

Let’s summarise all this:

x , x³ -2x  = 30 , comment

3 , 3³ - 2×3 = 21 , too small

4 , 4³ - 2×4 = 56 , too big

3.5 , 3.5³ - 2×3.5 = 35.875 , too big

3.4 , 3.4³ - 2×3.4 = 32.504 , too big

3.3 , 3.3³ - 2×3.3 = 29.337 , to small

3.35 , 3.35³ - 2×3.35 = 30.895375

x = 3.3

Take a look at this next example of using trial and improvement to solve to 2 decimal places.