Josh's solution for the equation [tex]\( x^2 - 6x - 7 = 0 \)[/tex] is incorrect. Let's examine why by solving the equation step-by-step and comparing it to Josh's solution:
Given the quadratic equation:
[tex]\[ x^2 - 6x - 7 = 0 \][/tex]
Step 1: Move the constant to the other side.
[tex]\[
x^2 - 6x = 7
\][/tex]
Step 2: Complete the square. To do this, we add and subtract [tex]\(\left( \frac{-6}{2} \right)^2 = 9\)[/tex] inside the equation:
[tex]\[
x^2 - 6x + 9 = 7 + 9
\][/tex]
[tex]\[
(x - 3)^2 = 16
\][/tex]
Step 3: Solve the resulting equation by taking the square root of both sides:
[tex]\[
x - 3 = \pm 4
\][/tex]
Step 4: Split into two separate equations:
[tex]\[
x - 3 = 4 \quad \text{or} \quad x - 3 = -4
\][/tex]
Step 5: Solve each equation for [tex]\(x\)[/tex]:
[tex]\[
x - 3 = 4 \quad \Rightarrow \quad x = 7
\][/tex]
[tex]\[
x - 3 = -4 \quad \Rightarrow \quad x = -1
\][/tex]
Therefore, the correct solutions to the equation [tex]\(x^2 - 6x - 7 = 0\)[/tex] are:
[tex]\[
x = 7 \quad \text{or} \quad x = -1
\][/tex]
Let's compare this with Josh's solution process:
Josh began correctly but made a critical mistake at the step where he solved [tex]\((x - 3)^2 = 16\)[/tex].
Instead of recognizing [tex]\(x - 3 = \pm 4\)[/tex], Josh incorrectly took each side:
[tex]\[
x - 3 = 16 \quad \text{and} \quad x - 3 = -16
\][/tex]
Which resulted in:
[tex]\[
x = 19 \quad \text{and} \quad x = -13
\][/tex]
These values are incorrect as solutions to the original equation. Therefore, Josh's solution is not correct. The correct solutions are [tex]\(x = 7\)[/tex] and [tex]\(x = -1\)[/tex].
