To solve the quadratic equation [tex]\(x^2 + 6x + 9 = 0\)[/tex] by factoring, follow these steps:
### Step 1: Write down the quadratic equation
The given quadratic equation is:
[tex]\[ x^2 + 6x + 9 = 0 \][/tex]
### Step 2: Factor the quadratic equation
To factor the quadratic equation, we look for two numbers that multiply to the constant term (9) and add up to the coefficient of the linear term (6).
The equation can be factored as:
[tex]\[ (x + 3)(x + 3) = 0 \][/tex]
This is because [tex]\( (x + 3) \times (x + 3) = x^2 + 6x + 9 \)[/tex].
### Step 3: Set each factor equal to zero and solve for [tex]\( x \)[/tex]
Set each factor equal to zero:
[tex]\[ x + 3 = 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ x = -3 \][/tex]
Since both factors are the same, [tex]\( x = -3 \)[/tex] is a repeated root. Therefore, the roots of the equation are:
[tex]\[ x = -3, -3 \][/tex]
### Step 4: Find the corresponding letter in the decoding table
The roots of our equation are [tex]\(-3, -3\)[/tex]. According to the decoder provided, we need to match these roots to find the relevant letter.
Looking at the decoder tables:
[tex]\[
\begin{tabular}{|l|l|l|l|l|}
\hline & & & & \\
\hline $-3,-4$ & 3,7 & $0,-6$ & $-3-4$ & 7,0 \\
\hline
\end{tabular}
\][/tex]
[tex]\[
\begin{tabular}{|l|l|l|l|l|l|}
\hline & & & & & \\
\hline $0, \frac{3}{2}$ & $0,-6$ & 6,1 & $4,-4$ & $-3,-4$ & 0,7 \\
\hline
\end{tabular}
\][/tex]
From the table, the correct decoding corresponding to the roots [tex]\(-3, -3\)[/tex] is:
[tex]\(\boxed{-3, -4}\)[/tex]
