This worksheet is about finding approximate solutions to more complex equations using trial and improvement.
We test a value which is near the solution and then home in to a more accurate solution until we reach the desired level of accuracy.
One solution to the following equation lies between x = 3 and x = 4.
x3 + 2x2 + 3x = 100
Using trial and improvement, find this solution to 1 d.p.
|x||x3 + 2x2 + 3x||Comment|
|3||33 + 2×32 + 3×3 = 54||too small|
|4||43 + 2×42 + 3×4= 108||too big|
|3.5||3.53 + 2×3.52 + 3×3.5= 77.875||too small|
|3.8||3.83 + 2×3.82 + 3×3.8= 95.152||too small|
|3.9||3.93 + 2×3.92 + 3×3.9= 101.439||just too big|
|3.85||3.853 + 2×3.852 + 3×3.85= 98.261625||just too small|
This shows that x = 3.85 is just too small but x = 3.9 is just too big, so the solutions lies between 3.85 and 3.9.
So to 1 decimal place, we can be sure that it will be x = 3.9 because anything above 3.85 will automatically round up to 3.9 anyway.
So x = 3.9 (to 1 d.p.)