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A Few Things You Should Learn Well in High School, But No One Told You – Part 4: The Square Root of x Squared
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Krystian Karczyński
Founder and General Manager of eTrapez.
Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.
He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.
Today, it’s going to be short. My quick question for you is:
What is the square root of x squared?
That is:
When I was still giving tutoring sessions, I asked this question to many high school and college students. Do you know how many knew the correct answer?
Two.
The breakdown of answers was roughly as follows:
75% answered (incorrect answer)
20% sensed the trick and were too scared to say anything
5% came up with some interesting and surprising things
I’m skipping the two students who knew the correct answer (they come from 7-8 years of intensive tutoring).
The Answer
I really don’t know why no one knows this, but the square root of x squared is the absolute value of x:
If it were x, that means if:
This means that if we take (-1) as x, we should get:
…but we don’t, we get:
You will encounter this trap more than once in college and need to be vigilant. I wasn’t asking my students this to conduct private surveys (during the time they were paying for), but because the issue kept coming up in various contexts.
Okay, but why? – An Example
For example, in the limits of functions (and sequences too), you often have to do this trick:
…and the standard mistake here is:
…which may or may not be true, because it should be:
…now we SHOULD check whether x is positive or negative (if we are in limits and we know, for example, that , we know it is positive, very positive).
Now using the definition of the absolute value:
…we know that if x is positive (well: non-negative), then:
…but if x is negative, then to our surprise we have:
The mistake obviously often affects the result and the final score on the exam.
So watch out for !
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