To transform the quadratic equation [tex]\( x^2 + 6x + 8 = 0 \)[/tex] into the form [tex]\((x - p)^2 = q\)[/tex], we need to complete the square. Here are the correct steps:
1. Start by moving the constant term to the right side of the equation:
[tex]\[ x^2 + 6x + 8 = 0 \][/tex]
[tex]\[ x^2 + 6x + 8 - 8 = 0 - 8 \][/tex]
[tex]\[ x^2 + 6x = -8 \][/tex]
2. Next, we add and subtract the square of half the coefficient of [tex]\(x\)[/tex] to complete the square. The coefficient of [tex]\(x\)[/tex] is 6, so half of it is 3, and its square is [tex]\(3^2 = 9\)[/tex]. Thus:
[tex]\[ x^2 + 6x + 9 - 9 = -8 \][/tex]
[tex]\[ (x + 3)^2 - 9 = -8 \][/tex]
3. Finally, simplify and isolate the squared term:
[tex]\[ (x + 3)^2 - 9 = -8 \][/tex]
[tex]\[ (x + 3)^2 = 1 \][/tex]
Thus, the correct transformation is:
[tex]\[ (x + 3)^2 = 1 \][/tex]
Reviewing the given options, the correct one is:
Step [tex]$1 \quad x^2 + 6x + 8 + 1 = 0 + 1$[/tex]
Step [tex]$2 \quad x^2 + 6x + 9 = 1$[/tex]
Step [tex]$3 \quad (x + 3)^2 = 1$[/tex]
So, the correct table is:
[tex]\[
\begin{aligned}
\text{Step 1} \quad & x^2 + 6x + 8 + 1 = 0 + 1 \\
\text{Step 2} \quad & x^2 + 6x + 9 = 1 \\
\text{Step 3} \quad & (x+3)^2 = 1 \\
\end{aligned}
\][/tex]
