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.
Answer
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.
For an easier trial and improvement example click here.
