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when I was asked by a friend to solve this problem 0.5x = sqrt(4-3x) + 2 I got the answers -4 and 0, but when he plugged them in he said that my values were incorrect, and that's true if he took the positive values of the root function, but if he considered the root to equal a negative value which some people and calculators call imaginary part my values would be correct.

So why does wolfram alpha and other calculators show no solution for this question although typing sqrt(4) into WolframAlpha shows 2 and -2.

adib
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    $\sqrt 2$ is typically understood as the positive branch of the square root, and $-\sqrt 2$ is the other branch. – abiessu Oct 22 '16 at 14:34
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    Well, WA does say “Result: 2” first, when you type “sqrt(4)”! (Later it displays all solutions to the equation $x^2=4$, which is not quite the same thing.) – Hans Lundmark Oct 22 '16 at 14:35
  • @abiessu but why in some cases people uses them as one function, I mean √4 should equal 2 or -2, it can be either of these values. – adib Oct 22 '16 at 14:36
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    By definition, it is taken to be +2. A function has to give the same value (and only one value) everytime you input the same number, otherwise it wouldn't be a function. – Hans Lundmark Oct 22 '16 at 14:37
  • but isn't considering the negative part of the square root value back in my function in the original question would be the right thing to do? – adib Oct 22 '16 at 14:39
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    See the first few paragraphs here for a concise summary: https://en.wikipedia.org/wiki/Square_root. The symbol $\sqrt{4-3x}$ in your equation has a very specific, generally agreed-on, meaning, namely the nonnegative real number whose square is $4-3x$. – Hans Lundmark Oct 22 '16 at 14:42
  • @HansLundmark What's confusing me now is that taking the negative square root would make the expression true – adib Oct 22 '16 at 14:50
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    Yes, but that's irrelevant, since the negative root is not what you have in the equation to begin with. The square root symbol denotes the nonnegative root (by universal agreement among mathematicians), so you just have to live with that. Because of this, $x=-4$ doesn't satisfy the equation: the left-hand side is $-2$ while the right-hand side is $+2$. – Hans Lundmark Oct 22 '16 at 14:56

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