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Equations and Inequalities

Equations and Inequalities

At a Glance - Keeping Both Solutions


Equations of the form x2 = (some positive number) have two solutions: the positive and the negative square roots of that positive number. That's okay—there's 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 y2 = 9.

We've got two options here: y can be either 3 or -3 because (-3)2 = 9 and 32 = 9. 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 x2 + 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's in the form of a word problem, sometimes you can toss the negative solution. If we're looking for the number of chickens Farmer Ben has let flee the coop, there's 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're 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 cm2. What is the length of one side of the square?


Exercise 1

How many solutions does the equation z2 = -25 have? If it has solutions, what are they? 


Exercise 2

How many solutions does the equation y2 – 3 = 13 have? If it has solutions, what are they? 


Exercise 3

How many solutions does the equation x2 = 0 have? If it has solutions, what are they?


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