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To solve the problem where \( f(a) = f^{-1}(a) \) for the function \( f(x) = 2x^2 - 3 \) defined for \( x \in \mathbb{R} \) and \( x < 0 \), we follow these steps:

1. **Find the inverse function \( f^{-1}(x) \):**
   Given \( y = f(x) = 2x^2 - 3 \),
   \[
   y + 3 = 2x^2
   \]
   \[
   x^2 = \frac{y + 3}{2}
   \]
   Since \( x < 0 \),
   \[
   x = -\sqrt{\frac{y + 3}{2}}
   \]
   Therefore,
   \[
   f^{-1}(x) = -\sqrt{\frac{x + 3}{2}}
   \]

2. **Set \( f(a) = f^{-1}(a) \):**
   \[
   2a^2 - 3 = -\sqrt{\frac{a + 3}{2}}
   \]

3. **Square both sides to eliminate the square root:**
   \[
   (2a^2 - 3)^2 = \left(-\sqrt{\frac{a + 3}{2}}\right)^2
   \]
   \[
   (2a^2 - 3)^2 = \frac{a + 3}{2}
   \]

4. **Expand and simplify the equation:**
   \[
   (2a^2 - 3)^2 = \frac{a + 3}{2}
   \]
   \[
   4a^4 - 12a^2 + 9 = \frac{a + 3}{2}
   \]
   \[
   8a^4 - 24a^2 + 18 = a + 3
   \]
   \[
   8a^4 - 24a^2 - a + 15 = 0
   \]

5. **Solve the polynomial equation:**
   This is a quartic equation and solving it exactly may require numerical methods or further factorization. However, we can check for possible rational roots.

6. **Check for possible rational roots using the Rational Root Theorem:**
   The Rational Root Theorem suggests possible roots could be \( \pm 1, \pm 3, \pm 5, \pm 15 \). By testing these values, we can identify any real roots.

   For \( a = -1 \):
   \[
   8(-1)^4 - 24(-1)^2 - (-1) + 15 = 8 - 24 + 1 + 15 = 0
   \]
   So, \( a = -1 \) is a solution.

Therefore, \( a = -1 \) is the value where \( f(a) = f^{-1}(a) \).

Would you like more details on solving quartic equations, or do you have any other questions?

Here are some related questions:
1. How can we verify that \( f(x) \) and \( f^{-1}(x) \) are indeed inverse functions?
2. What are the graphical interpretations of the functions \( f(x) \) and \( f^{-1}(x) \)?
3. How do we derive inverse functions for more complex functions?
4. What is the significance of the domain restriction \( x < 0 \)?
5. How do we solve quartic equations in general?
6. Can there be more than one solution to \( f(a) = f^{-1}(a) \)?
7. What are some numerical methods for solving polynomial equations?
8. How do we factorize quartic equations efficiently?

**Tip:** When solving equations involving both functions and their inverses, always check the domain restrictions and ensure your solutions fit within those domains.

Learn how to solve the problem where f(a) = f^-1(a) for the function f(x) = 2x^2 - 3. Detailed step-by-step solution involving finding the inverse function, setting equations, and solving polynomial equations. Explore the rational root theorem and its application. Suitable for advanced high school students.

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