Equations of the form *x*^{2} = (some positive number) have two solutions: the positive and the negative square roots of that positive number. That's okay—there is room here for a second answer. The more, the merrier.

...what, of course we're not saying that through gritted teeth.

### Sample Problem

Solve the equation *y*^{2} = 9.

*y* can be either 3 or - 3. Since mathematicians like to abbreviate some things (winner of the Understatement of the Month Award), we write these two answers together as

*y* = ± 3.

### Sample Problem

Solve the equation *x*^{2} + *x* - 4 = *x*.

Here's the work:

**Be Careful:** Sometimes a problem is set up so that a negative answer doesn't make sense. When there are numbers involved, anything goes, but when it is in the form of a word problem, sometimes you can toss the negative solution. If we are looking for the number of chickens Farmer Ben has let flee the coop, there is no sense in giving him a number that includes negative chickens.

In these cases, only give the positive root as your final answer, but point out that this is what you are doing. This will convince your teacher and your cute study partner that you know what you're talking about. We probably wouldn't mention the chickens, though.

## Keeping Both Solutions Practice:

A square has an area of 100 cm^{2}. What is the length of one side of the square? | |

Draw a square. Tracing a pad of Post-It notes works like a charm if your freehand lines are a little shaky. Label the side length *s* (or *x*, or *y*, or @, or whatever variable you like). We can see that the area *A* of the square is given by the formula *A* = *s*^{2}.
Because *A* is 100, we find the answers if we solve the formula. 100 = *s*^{2} *s* = ± 10
However, the answer -10 doesn't make any sense in the context of this problem. What are negative centimeters? The metric system may seem a little weird to many of us, but it's not *that* weird. A good way to show work here would be to write something like this sentence: "Since the length of a side can't be negative, we can eliminate the answer *s* = -10. The final answer is *s* = 10 cm." Then sit back, cross your arms, and wait for the extra credit and accolades to come rolling in. | |

How many solutions does the equation *z*^{2} = -25 have? If it has solutions, what are they?

Answer

There are no solutions, since any real number squared is positive.

How many solutions does the equation *y*^{2} - 3 = 13 have? If it has solutions, what are they?

Answer

There are 2 solutions: *y* = ± 4.

How many solutions does the equation *x*^{2} = 0 have? If it has solutions, what are they?

Answer

There is 1 solution: *x* = 0.