To find the solutions to the quadratic equation \( x^2 + 6x + 9 = 0 \) using the quadratic formula, we can identify the coefficients:
- \( a = 1 \)
- \( b = 6 \)
- \( c = 9 \)
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Now, we can plug in the values of \( a \), \( b \), and \( c \):
- First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 6^2 - 4 \cdot 1 \cdot 9 = 36 - 36 = 0 \]
-
Since the discriminant is 0, we will have exactly one real solution.
-
Now, we can substitute the values into the quadratic formula:
\[ x = \frac{-6 \pm \sqrt{0}}{2 \cdot 1} = \frac{-6 \pm 0}{2} = \frac{-6}{2} = -3 \]
Thus, the solution to the equation \( x^2 + 6x + 9 = 0 \) is
\[ \boxed{-3} \]
This means there is a double root at \( x = -3 \).