Common Types of Mistakes: Omitted Solutions
Sometimes only one of two or more solutions to an equation are found and the other solution(s) are omitted from an answer. Take care especially with square roots.
Example 1: Solve
Mistake:
Click to see the mistake in red. Click to remove the red highlighting.
This equation has not one, but two solutions, one of which is 4. However, there's another number whose square is 16:
Click to see a correct solution. Click to hide the correct solution.
Remember that the square root symbol √ denotes just the positive square root, so you have to explicitly indicate a negative square root.
If you have studied trigonometric functions, then Example 2 reminds you to be careful when solving equations involving these periodic functions.
Example 2: Solve
Mistake:
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This equation has not one, but an infinite number of solutions:
Click to see a correct solution. Click to hide the correct solution.
(for k = ...,-3,-2,-1,0,1,2,3,...)
There are two problems with the mistake. The first is that there are two angles in a full period of sine for which the value of sine is 1/2. The second is that the sine function has period 2π = 360°, and so integer multiples of 2π = 360° must be added to the two solutions that lie in one period to get the full set of solutions.
If your first step in solving this equation is to write x=sin-1(1/2), then you may pick up only the solution in the first quadrant. If you also correctly regard sin-1(1/2) as the reference angle for the solution in the second quadrant (where sine is also positive), you will pick up the second solution. Otherwise you will miss the second solution that lies in the first period of sine. You also risk missing this second solution if you rely on your calculator to compute the value of the inverse sine, unless you think carefully about the properties of the sine function and understand what the inverse sine function tells you.
Make sure you find all solutions for an equation. If you use a calculator to help find the solutions of an equation, think about whether your calculator is telling you everything you need to know.