solve −3x^2−36x−108 using the quadratic formula

To solve the quadratic equation \(-3x^2 - 36x - 108 = 0\) using the quadratic formula, follow these steps:

The quadratic formula is:

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

### Step 1: Identify the coefficients
In the given equation \(-3x^2 - 36x - 108 = 0\), we have:
- \(a = -3\)
- \(b = -36\)
- \(c = -108\)

### Step 2: Plug the values into the quadratic formula

\[
x = \frac{-(-36) \pm \sqrt{(-36)^2 - 4(-3)(-108)}}{2(-3)}
\]

\[
x = \frac{36 \pm \sqrt{1296 - 1296}}{-6}
\]

### Step 3: Simplify the discriminant

\[
\sqrt{1296 - 1296} = \sqrt{0} = 0
\]

### Step 4: Solve for \(x\)

\[
x = \frac{36 \pm 0}{-6} = \frac{36}{-6}
\]

\[
x = -6
\]

### Conclusion:
The solution to the equation \(-3x^2 - 36x - 108 = 0\) is:

\[
x = -6
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How is the discriminant used in solving quadratic equations?
2. What if the discriminant is negative, how does that affect the solution?
3. How can you factor this quadratic equation instead of using the quadratic formula?
4. What does it mean when the discriminant is zero?
5. How does changing the sign of \(a\), \(b\), or \(c\) affect the solution?

**Tip**: The discriminant (\(b^2 - 4ac\)) determines the number and nature of the roots—positive means two distinct solutions, zero means one solution, and negative means complex solutions.

This solution shows how to solve the quadratic equation -3x^2 - 36x - 108 using the quadratic formula. It includes detailed steps, explains the discriminant, and provides insights into quadratic equation solving techniques. Suitable for Grades 9-10.

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