Let's carefully analyze each step of Tyrianne's work to identify the error.
1. Original Equation:
[tex]\[
\frac{1}{2}(x+4)^2 - 3 = 29
\][/tex]
2. Isolating the squared term:
[tex]\[
\frac{1}{2}(x+4)^2 = 29 + 3
\][/tex]
[tex]\[
\frac{1}{2}(x+4)^2 = 32
\][/tex]
So far, these steps are correct.
3. Removing the fraction by multiplying both sides by 2:
[tex]\[
(x+4)^2 = 32 \times 2
\][/tex]
[tex]\[
(x+4)^2 = 64
\][/tex]
This step is correct. However, in Tyrianne's work, she did:
[tex]\[
(x+4)^2 = 16
\][/tex]
This is incorrect. The correct value should be 64, not 16.
4. Taking the square root of both sides:
[tex]\[
x+4 = \pm \sqrt{64}
\][/tex]
[tex]\[
x+4 = \pm 8
\][/tex]
This should follow correctly if Step 3 was done properly.
5. Solving for x:
[tex]\[
x = -4 + 8 \quad \text{or} \quad x = -4 - 8
\][/tex]
[tex]\[
x = 4 \quad \text{or} \quad x = -12
\][/tex]
These solutions would be correct if all steps were done correctly.
Therefore, the error occurred in Step 2 when she incorrectly solved:
[tex]\[
(x+4)^2 = 16
\][/tex]
Instead of correctly solving as:
[tex]\[
(x+4)^2 = 64
\][/tex]
Thus, Tyrianne made an error in Step 2.
