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"There exist a,b ∈ ℝ such that {x ∈ ℝ | ax^2 = b} = ∅"

I am not quite sure where to start. I do see many sets of a,b,x that satisfies ax^2 = b, but I am not sure what it means by it is equal to an empty set, ∅.

Am I completely on the wrong track? Help would be very much appreciated!

Isaac Seo
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    The empty set means that there are no values of $x$ that solve the equation $ax^2=b$. Now if $a=b=1$ then there indeed are values of $x$ that solve $x^2=1$: $$ {x \in \mathbb{R} : x^2=1} = {-1,1}$$ Is it solvable for all $a,b \in \mathbb{R}$? – Michael Oct 08 '19 at 18:21
  • Note that $x^2$ can never be negative. – amsmath Oct 08 '19 at 18:24
  • @Michael Aha, it's more clear now. What about a = b = 0 then? Will this make it:

    { ∈ ℝ : 0 = 0 } = { } ? I'm not sure if this is correct reasoning.

     – Isaac Seo Oct 08 '19 at 18:48
  • When $a=b=0$, then all $x$ satisfy the condition, ${x \in \mathbb{R} : 0x^2=0}=\mathbb{R} \neq {}$. What about $a=1$, $b=-1$? (The set in question is the empty set if no real number $x$ exists that fullfils the condition) – DavidP Oct 08 '19 at 19:12
  • I understand now. I was misunderstanding the statement {x ∈ ℝ | ax^2 = b} = ∅. Thank you for helping! – Isaac Seo Oct 09 '19 at 00:11

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