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November 4th, 2017, 09:41 AM   #1
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Wrong Answer for Correct Procedure?

Here is a problem:

Solve: (x^2 - 4) = x^2 - 10x + 16

Procedure 1:

factor both sides for (x+2)(x-2) = (x-8 )(x-2)
divide out common factors (x+2) = (x -8 )
simplify 0 = -10
which is obviously wrong.


Procedure 2:
subtract all values on left from both sides for 0= x^2 - 10x + 16 - x^2 + 4
simplify 0= -10x + 20
which means 10x=20
x=2


I don't understand why is the first procedure wrong while the second procedure is correct?

Last edited by skipjack; November 4th, 2017 at 09:44 AM.
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November 4th, 2017, 09:47 AM   #2
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In the first method, dividing by x-2 isn't possible if x = 2 (because division by zero isn't defined). Hence, you should consider whether x = 2 is a solution.
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November 4th, 2017, 09:54 AM   #3
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Quote:
Originally Posted by Dynamics View Post
Here is a problem:

Solve: (x^2 - 4) = x^2 - 10x + 16

Procedure 1:

factor both sides for (x+2)(x-2) = (x-8 )(x-2)
divide out common facotrs (x+2) = (x -8 )
simplify 0 = -10
which is obviously wrong
If x = 2 then you get 0 = -10. So this procedure is only valid if $\displaystyle x \neq 2$. (The reason for this is that if x = 2 then you are dividing both sides by x - 2 = 2 - 2 = 0, which is, of course, impossible.)

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Originally Posted by Dynamics View Post
Procedure 2:
subtract all values on left from both sides for 0= x^2 - 10x + 16 - x^2 + 4
simplify 0= -10x + 20
which means 10x=20
x=2
As this procedure shows x = 2, which your first method shows is incorrect, you have a problem. (Always look for this sort of thing.)

It's almost done. Can you finish?

-Dan
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November 4th, 2017, 10:15 AM   #4
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Hi! Thank you for your help!

I think I understand your point. Because we cannot divide by zero, we know that dividing by some factor (x-n) is only valid if x does not equal n, (and, conversely, that dividing by some factor (x+n) is only valid if -x does not equal n). Therefore, we know that dividing by some factor (x-n) is valid for all x values except for x=n. Therefore, if we find that dividing by factor (x-n) renders an invalid result, or a contradiction, then x must be the only number which we understood to be that which would make the division of (x-n) to be invalid- namely, n. So, in a sense, dividing by 0 in the form of dividing by a factor (x-n) can give you the correct answer (in a negative way) since it reveals the only number that could have led to such a contradictory result.
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November 6th, 2017, 10:39 AM   #5
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In general, you have to be careful anytime you divide both sides of an equation by something involving the "unknown", x, say. Once you have arrived at one or more values for x, check to be sure that value of x does not make the divisor 0.
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