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The given function is:

\[
f(x) = \frac{3}{2}x^2 + \frac{5}{3}x - \frac{190}{27}
\]

We are tasked with finding the set of solutions for \( f(x) = 0 \), which means solving the quadratic equation:

\[
\frac{3}{2}x^2 + \frac{5}{3}x - \frac{190}{27} = 0
\]

Let me solve this quadratic equation step by step.The solutions to the equation \( f(x) = 0 \) are approximately:

\[
x_1 \approx -2.79 \quad \text{and} \quad x_2 \approx 1.68
\]

Therefore, the solution set is approximately \( \{-2.79, 1.68\} \).

Would you like further details or have any questions?

Here are five related questions you might find useful:
1. How do we derive the exact form of the roots for a quadratic equation?
2. What is the significance of solving \( f(x) = 0 \) in mathematical contexts?
3. How can we confirm whether the solutions are correct?
4. How does the discriminant determine the number of solutions for a quadratic equation?
5. What methods exist for solving quadratic equations other than factoring or the quadratic formula?

**Tip:** You can always check the discriminant (\(b^2 - 4ac\)) to predict the nature of the solutions in quadratic equations.

Soit f la fonction définie sur R par : f : x -> (3/2)x^2 + (5/3)x - 190/27. Quel est l'ensemble des solutions de f(x) = 0 sur R ?

Learn how to solve the quadratic equation f(x) = (3/2)x² + (5/3)x - 190/27. This solution involves using the quadratic formula to find the roots, with step-by-step explanations suitable for high school students (Grades 9-12).

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