To solve the quadratic equation 

\[
x^2 + 12x + 27 = 0,
\]

we can use the quadratic formula:

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

where \(a = 1\), \(b = 12\), and \(c = 27\). Let's substitute these values into the formula:

1. Calculate the discriminant (\(b^2 - 4ac\)):

   \[
   b^2 - 4ac = 12^2 - 4 \cdot 1 \cdot 27 = 144 - 108 = 36.
   \]

2. Find the values of \(x\):

   \[
   x = \frac{-12 \pm \sqrt{36}}{2 \cdot 1} = \frac{-12 \pm 6}{2}.
   \]

3. Solve for the two possible values of \(x\):

   - For the positive root:

     \[
     x = \frac{-12 + 6}{2} = \frac{-6}{2} = -3.
     \]

   - For the negative root:

     \[
     x = \frac{-12 - 6}{2} = \frac{-18}{2} = -9.
     \]

So, the solutions are:

\[
x = -3 \quad \text{and} \quad x = -9.
\]

Would you like more details or have any questions?

### Related Questions:

1. How do you complete the square to solve this equation?
2. What is the axis of symmetry for the parabola defined by this equation?
3. Can the quadratic formula be used for all quadratic equations?
4. How can you graph the equation \(x^2 + 12x + 27 = 0\)?
5. What are the sum and product of the roots of this equation?

**Tip:** The discriminant (\(b^2 - 4ac\)) helps determine the nature of the roots: if positive, the roots are real and distinct; if zero, they are real and equal; and if negative, they are complex.

Learn how to solve the quadratic equation x^2 + 12x + 27 = 0 using the quadratic formula. This step-by-step guide explores key algebraic concepts, including solving for the roots and calculating the discriminant. Perfect for students in Grades 8-10.

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