We are given the equation:

\[
0 = -2x^2 - 7x - 3
\]

To solve this quadratic equation, we can either factor it (if factorable) or use the quadratic formula. Since factoring might be complex here, I'll proceed with the quadratic formula.

The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( 0 = -2x^2 - 7x - 3 \), we have:

- \( a = -2 \)
- \( b = -7 \)
- \( c = -3 \)

### Step 1: Calculate the discriminant

The discriminant \( \Delta \) is given by:

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

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

\[
\Delta = (-7)^2 - 4(-2)(-3) = 49 - 24 = 25
\]

### Step 2: Apply the quadratic formula

Now we use the quadratic formula:

\[
x = \frac{-(-7) \pm \sqrt{25}}{2(-2)} = \frac{7 \pm 5}{-4}
\]

This gives two possible solutions:

1. \( x_1 = \frac{7 + 5}{-4} = \frac{12}{-4} = -3 \)
2. \( x_2 = \frac{7 - 5}{-4} = \frac{2}{-4} = -\frac{1}{2} \)

### Final solutions:

\[
x_1 = -3 \quad \text{and} \quad x_2 = -\frac{1}{2}
\]

Would you like more details or have any questions about these steps?

Here are 5 related questions to deepen your understanding:
1. What is the discriminant, and how does it affect the nature of the roots?
2. How can the quadratic formula be derived from completing the square?
3. Can you graph the quadratic function \( y = -2x^2 - 7x - 3 \) and find its vertex?
4. How would the solutions change if the quadratic equation was \( 0 = 2x^2 + 7x + 3 \)?
5. What are the sum and product of the roots of this quadratic equation?

**Tip:** Always check the discriminant first—it tells you if the roots are real, rational, or complex!

Learn how to solve the quadratic equation 0 = -2x^2 - 7x - 3 using the quadratic formula. This problem explores key algebraic concepts, including calculating the discriminant, applying the quadratic formula, and analyzing the nature of the roots. Suitable for students in Grades 9-12.

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