# Equations and Inequalities

### Topics

## Introduction to :

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.

#### Example 1

A square has an area of 100 cm |

#### Exercise 1

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

#### Exercise 2

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

#### Exercise 3

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