High School

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------------------------------------------------ This is Josh's solution for the equation [tex]x^2 - 6x - 7 = 0[/tex]:

[tex]
\begin{aligned}
x^2 - 6x - 7 & = 0 \\
x^2 - 6x & = 7 \\
x^2 - 6x + 9 & = 7 + 9 \\
(x - 3)^2 & = 16 \\
x - 3 & = 16 \\
x & = 19 \\
x - 3 & = -16 \\
x & = -13
\end{aligned}
[/tex]

Is Josh's solution correct? Explain.

Answer:

Answer :

Josh's solution for the equation [tex]\(x^2 - 6x - 7 = 0\)[/tex] appears to have an error. Let's go through the steps to identify what went wrong and find the correct solution.

1. Original Equation:
[tex]\[
x^2 - 6x - 7 = 0
\][/tex]

2. Josh's Incomplete Step:
[tex]\[
x^2 - 6x = 7
\][/tex]

3. Completing the Square:
Josh correctly added 9 to both sides to complete the square, but there was a mistake in the subsequent steps.
[tex]\[
x^2 - 6x + 9 = 7 + 9
\][/tex]
[tex]\[
(x-3)^2 = 16
\][/tex]

4. Solving for [tex]\(x\)[/tex]:
Here, Josh made an error. The correct approach is:
[tex]\[
x - 3 = \sqrt{16} \quad \text{or} \quad x - 3 = -\sqrt{16}
\][/tex]

This results in:
[tex]\[
x - 3 = 4 \quad \Rightarrow \quad x = 7
\][/tex]
[tex]\[
x - 3 = -4 \quad \Rightarrow \quad x = -1
\][/tex]

Josh's solution incorrectly concluded with [tex]\(x = 19\)[/tex] and [tex]\(x = -13\)[/tex], which are not correct for this equation.

Correct Solutions:
- The values of [tex]\(x\)[/tex] are 7 and -1.

Therefore, Josh's solution is not correct, and the correct solutions for the equation [tex]\(x^2 - 6x - 7 = 0\)[/tex] are [tex]\(x = 7\)[/tex] and [tex]\(x = -1\)[/tex].