Question

using the quadratic formula, find the solution to x^2 + 6x + 9 = 0.

Answer 1

To solve the quadratic equation \(x^2 + 6x + 9 = 0\) using the quadratic formula, we start by identifying the coefficients \(a\), \(b\), and \(c\) in the standard form \(ax^2 + bx + c = 0\).

Here:
- \(a = 1\)
- \(b = 6\)
- \(c = 9\)

The quadratic formula is given by:

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

First, we calculate the discriminant (\(b^2 - 4ac\)):

\[
b^2 = 6^2 = 36
\]
\[
4ac = 4 \cdot 1 \cdot 9 = 36
\]
\[
b^2 - 4ac = 36 - 36 = 0
\]

Since the discriminant is 0, there is one real solution (a repeated root). We now substitute the values into the quadratic formula:

\[
x = \frac{{-6 \pm \sqrt{0}}}{2 \cdot 1}
\]
\[
x = \frac{{-6 \pm 0}}{2}
\]
\[
x = \frac{{-6}}{2} = -3
\]

Thus, the solution to the equation \(x^2 + 6x + 9 = 0\) is:

\[
\boxed{-3}
\]