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irrational function

So, as you can see in the photo above, I solved this equation and I got 2 results as an answer. Both of which seem correct to me. But my teacher stated that the -2 solution is incorrect, and that 1 is the only correct answer. I graphed the function, in a software, and I saw that the function really does only cut the x axis at 1 and that the function is only defined at x greater or equal to 2. With all of that being stated, what I don't understand is: why -2 isn't a solution and why the function is only defined at x greater or equal to 2.

P.S. In case that you are wondering, Solución means solution in Spanish.

Marco Trevino
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    If $x=-2$, then first note that $\sqrt {2-x}=\sqrt 4 = 2$. Thus, the left hand would evaluate to $2-(-2)=4$ which is not $0$. – lulu Sep 26 '21 at 13:48
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    But...it is not at all true that the left hand is only defined for $x≥2$. On its face (that is, assuming there are no restrictions on $x$ you did not mention) it would appear to be defined for $x≤2$. Indeed, $x=1$ is not in the range $x≥2$. – lulu Sep 26 '21 at 13:50

2 Answers2

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I think the key idea here is to be careful when squaring both sides of an equation.

Any solution for an equation $L(x) = R(x)$ is surely also a solution for $(L(x))^2$ = $(R(x))^2$, but the reverse is not necessarily true.

If $x$ satisfies $(L(x))^2 = (R(x))^2$ then we can only deduce that $|L(x)| = |R(x)|$, but after taking away the absolute value signs around each side, the equality might not hold (the signs of either side may not match).

You found 2 candidate solutions from the squared equation but you need to check them in the original equation to make sure they still work on the "un-squared expressions".

bluebird
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Put $-2$ in the original equation. You can see the result is not $0$. That one extra root came because you squared the equation.

Zootopia
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    As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community Sep 26 '21 at 14:03
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    I wonder if there is a minimal character number that can cause an automated message from "Community"? What you've written is definitely NOT unclear for the context of the question. – Dave L. Renfro Sep 26 '21 at 14:35