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So I had to solve this equation in my math book, $3x=\sqrt{18x + 72}$ the solutions I worked out are $x= -2$ and $x= 4$.

Checking the validity of the solutions, for $x= 4$ gives $12=12$ and for $x= -2$ gives $-6=\sqrt{36}$

My correction book stated that $x= -2$ is not a valid solution, however the $\sqrt{36}$ is $\pm6$, so I was wondering why it is not correct? Am I wrong or is there an error in my book?

Is there a valid reasoning as to why the proposed solution is not correct?

My math is not in english and I am also not a native english speaker so please keep that in mind. (sorry if I'm using the wrong formatting, this is my first post here)

cpiegore
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BroodjeRijst
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    There's a difference between the equations $3x=\sqrt{18x+72}$ and $(3x)^2=18x+72$. The second one has $x=-2$ as a solution but the first one does not. By definition, the square-root function always gives a nonnegative value, so $\sqrt{18x+72}$ can never be negative, regardless of the value of $x$. – Greg Martin Aug 09 '22 at 16:54
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    The book is right. $-6\ne \sqrt{36}.$ When we use the $\sqrt{}$ symbol, we always mean the principal (positive) root. When you square both sides to solve an equation there is always a risk of introducing artificial solutions and all solutions should be checked. – user317176 Aug 09 '22 at 16:55
  • Please use MathJax. See https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Paul Frost Aug 09 '22 at 16:58

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