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When does a function: $f(x)=3ax^2 + 2bx + c $ has no solution?

can i say that if the discriminant $4b^2 - 12ac$ is a negative number because according to the quadratic root formula square root portion is a negative number and square root of a negative number is undefined

insipidintegrator
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user307640
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    Is it 2bx or 2bc? If yes, then you can. – insipidintegrator Oct 06 '22 at 13:06
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    What do you mean by "no solution"? Every quadratic polynomial has a complex root, i.e., an $x$ such that $f(x)=0$. Of course, for $a=b=0$, it is not a quadratic polynomial, and $c$ need not be zero. – Dietrich Burde Oct 06 '22 at 13:08
  • https://math.stackexchange.com/questions/2964339/i-don-t-understand-this-can-you-explain-it-simply – insipidintegrator Oct 06 '22 at 13:12
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    Your answer is almost correct ("almost" = you forgot to study the case $a=0$), provided you change your question (and title) to: When does a function: $f(x)=3ax^2 + 2bx + c $ have no real zero? – Anne Bauval Oct 06 '22 at 13:17
  • @insipidintegrator did I miss something or did you put a wrong link? – Anne Bauval Oct 06 '22 at 13:18
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    You speak of a solution, but what you give is no equation, it is only a function. You can either ask when $f(x)$ has no (real) root or when $f(x)=0$ has no (real) solution. – Peter Oct 06 '22 at 13:26
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    The second answer by PrincessEevee deals with a similar problem. @AnneBauval – insipidintegrator Oct 06 '22 at 14:13

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Your approach is correct. When $4b^2-12 ac<0$, function $f$ has two complex roots (nonzero imaginary part). When $4b^2-12 ac \ge 0$ it has one or two distinct real roots.

PierreCarre
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