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].
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].
