Let's carefully go through the steps Montiah performed to solve the quadratic equation [tex]\(3(x+8)^2 + 8 = 83\)[/tex].
1. Starting from the given equation [tex]\(3(x+8)^2 + 8 = 83\)[/tex]:
[tex]\[ 3(x+8)^2 + 8 = 83 \][/tex]
First, subtract 8 from both sides to isolate the quadratic term:
[tex]\[ 3(x+8)^2 + 8 - 8 = 83 - 8 \][/tex]
[tex]\[ 3(x+8)^2 = 75 \][/tex]
Step 1 is correct.
2. Next, divide both sides by 3 to solve for [tex]\((x+8)^2\)[/tex]:
[tex]\[ \frac{3(x+8)^2}{3} = \frac{75}{3} \][/tex]
[tex]\[ (x+8)^2 = 25 \][/tex]
In Step 2, Montiah has made an error. Instead of [tex]\((x+8)^2 = 25\)[/tex], she wrote [tex]\((x+8)^2 = 225\)[/tex].
3. If [tex]\((x+8)^2 = 25\)[/tex], to solve for [tex]\(x\)[/tex], take the square root of both sides:
[tex]\[ x+8 = \pm 5 \][/tex]
Step 3 is correct according to the corrected version of Step 2.
4. Solving [tex]\(x+8 = \pm 5\)[/tex]:
[tex]\[ x+8 = 5 \quad \text{or} \quad x+8 = -5 \][/tex]
[tex]\[ x = 5 - 8 \quad \text{or} \quad x = -5 - 8 \][/tex]
[tex]\[ x = -3 \quad \text{or} \quad x = -13 \][/tex]
Step 4 is correct according to the corrected version of Step 2.
Hence, Montiah made an error in Step 2.
