Algebra 1 Variables as Roots
Math-Prof HOME Algebra 1 Table of Contents Ask A Question PREV NEXT

Roots
Variables as Roots

 

 Once upon a time

 you might have learned about things called roots.
 
 Maybe you called it a radical.
 
 Maybe you just called it a rad.
 
 Whatever you called it,
 it looked like this ...
 

 
 It doesn't mean much by itself.
 It needs to have something inside of it ...
 

 
 All together, the number and the squiggle means:
 

 THE NUMBER THAT TIMES ITSELF EQUALS 4

 
 That is ...
 

 
 Now we just have to figure out
 what number times itself equals 4.
 
 This one us not too tough.
 Most people know that ...

 2 x 2 = 4

 
 That means:
 

 
 (At the end of this page, we will talk about another answer to this problem.
 To be completely correct, we should mention this answer now
 Later on in Algebra, we will use this other answer in some problems)
 
 OK, but what's this got to do with X?
 
 Thanks, I was hoping that you would ask ...
 What if you have this problem ...
 

 X2 = 25

 
 How do you turn this puppy into:
 

 X = SOMETHING

 
 We can't get there by adding or subtracting anything.
 We can't get there by multiplying or dividing by anything.
 What now?
 

 
 So what we do here, 
 is to drop one of those root things onto each side of the = .
 Like this ...

 X2 = 25

 

 
 So what have we got?
 
 What does this mean?
 

 
 It means the number that times itself equals 25.
 So what number times itself equals 25?
 

 5 5 = 25

 
 So:
 

 
 That makes our problem ...
 

 
 Now lets look at the messy X part.
 What does this mean?
 

 
 It means ...
 

THE THING THAT TIMES ITSELF EQUALS X2

 
 So what times itself equals X 2?
 

 X X = X2

 
 So:
 

 
 That makes our problem ...
 

 

 
 OK, here's that little detail I mentioned earlier.
 
 The extra answer you get in these problems
 works like this.
 
 When we said
 "What times itself equals 4?"
 
 We gave the answer as 2.
 That's true. But there is another number that times itself equals 4.
 
 The other answer is -2.
 
 A negative times a negative is a positive.
 So -2 x -2 = 4 also.
 
 Later on, we will use this "negative root" in problems.
 For now, we will mostly ignore it.
 
 The positive root that we will use,
 is most often called the principal square root.
 

   copyright 2005 Bruce Kirkpatrick

Math-Prof HOME Algebra 1 Table of Contents Ask A Question PREV NEXT