# Solving an equation

Hi, I'm stuck on the following implicit function question.

Q. Find the values of t_1, t_2 such that every solution theta is determined as a C^1 function of t_1 and t_2.

The equation is $\theta ^3 + t_1 \theta + t_2 = 0$.

Ok from what I gather we want $\theta = g\left( {t_1 ,t_2 } \right)$. This one is a little different to the others I've done before and I think one of the conditions I need to check, for this particular case is:

$$\frac{{\partial F}}{{\partial \theta }} = 3\theta ^2 + t_1 \ne 0,F\left( {t_1 ,t_2 ,\theta } \right) = \theta ^3 + t_1 \theta + t_2$$ where the not equal to zero condition is satisified at a specific point (..,..,..).

Of course if I want to apply the implicit function theorem I need a specific point (t_1,t_2), maybe call it (x,y) to avoid ambiguity. Usually I'm given
one but in this case I'm not so I'm at a loss as to what I need to do. I was told that this question requires a bit of thought but I'm too stupid to work this out so can someone help me out? Any help would be good thanks.

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Using the implicit function theorem it came down to $$\frac{{\partial F}}{{\partial \theta }} = 3\theta ^2 + t_1 \ne 0$$.
But like I said before I'm not given a specific point to work with as is usually the case. The above is what I obtained and the answer is that C^1 solutions exist when $4t_1 ^3 + 27t_2 ^2 \ne 0$. I cannot see how they came to that conclusion and I don't understand how you could go any further than what I have given as my answer. Can someone please help me out?