my question is with regards to two different problems (both containing a statement A and statement B) that are quite similar. The objective is to decide how the implication arrow is supposed to be pointed (->, <-, <-> or not at all):
A: $\sqrt{10-x^2}=3x$, B: $x=1$
A: $\sqrt{10-x^2}=3x$, B: $x^2=1$
For both problems I have the dilemma of how to take care of the second root. That is, the square root of 9 would mean having an answer of +/-3 and how would that work in conjunction with the implication arrow?
Any help is very appreciated, thanks!
Suppose we square both sides of (A). We then have
$$ 10-x^2 = 9x^2 $$
From this, we can conclude that $10x^2 = 10$, or $x^2 = 1$. That is to say, candidates for $x$ are $1$ and $-1$. But if we substitute these values back into (A), we find that only $x = 1$ works; $x = -1$ yields $\sqrt{10-(-1)^2} = \sqrt{9} = 3 \not= 3(-1)$. (As André Nicolas writes in his comment, we usually define $\sqrt{x}$ to be the positive square root, by convention.)
Therefore, $\sqrt{10-x^2} = 3x$ is equivalent to $x = 1$, and implies $x^2 = 1$, but $x^2 = 1$ does not imply $\sqrt{10-x^2} = 3x$.