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We are given the quadratic equation:

\[
-3x^2 - x - 1 = 0
\]

To compute the value of the discriminant and determine the number of real solutions, we will use the quadratic formula, where the general form of a quadratic equation is:

\[
ax^2 + bx + c = 0
\]

For our equation:
- \( a = -3 \)
- \( b = -1 \)
- \( c = -1 \)

### Discriminant Formula:
The discriminant \( \Delta \) of a quadratic equation is given by:

\[
\Delta = b^2 - 4ac
\]

Substituting the values of \( a \), \( b \), and \( c \):

\[
\Delta = (-1)^2 - 4(-3)(-1) = 1 - 12 = -11
\]

### Conclusion:
- The discriminant is \( \Delta = -11 \).
- Since the discriminant is negative, the quadratic equation has **no real solutions** but **two complex solutions**.

Would you like more details on the solution or clarification on any steps?

Here are some related questions:
1. How do discriminants affect the type of solutions in a quadratic equation?
2. What is the complete quadratic formula including both real and complex solutions?
3. Can complex solutions be visualized in some way?
4. How would changing the coefficients affect the discriminant?
5. What are the conditions for a quadratic equation to have exactly one real solution?

**Tip**: A negative discriminant always leads to complex (non-real) solutions for quadratic equations.

Compute the value of the discriminant and give the number of real solutions of the quadratic equation -3x^2 - x - 1 = 0.

Learn how to compute the discriminant for the quadratic equation -3x^2 - x - 1 = 0 and determine the number of real solutions. This problem focuses on using the discriminant formula to classify the solutions of the equation. Suitable for Grades 9-10.

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