Solving the Quadratic: Tackling (x-5)² - 5 = -12 + 2x

To solve the equation $(x-5)^2 - 5 = -12 + 2x$, follow these steps:

• Step 1: Expand the square on the left side of the equation:
  $(x-5)^2 = x^2 - 10x + 25$
• Step 2: Substitute this back into the equation:
  $x^2 - 10x + 25 - 5 = -12 + 2x$
• Step 3: Simplify the equation:
  $x^2 - 10x + 20 = -12 + 2x$
• Step 4: Rearrange the equation by moving all terms to one side:
  $x^2 - 10x + 20 - 2x + 12 = 0$
  This simplifies to $x^2 - 12x + 32 = 0$.
• Step 5: Use the Quadratic Formula, where $a = 1$, $b = -12$, and $c = 32$:
  $x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \times 1 \times 32}}{2 \times 1}$
• Step 6: Calculate the discriminant and simplify:
  $x = \frac{12 \pm \sqrt{144 - 128}}{2}$
  $x = \frac{12 \pm \sqrt{16}}{2}$
  $x = \frac{12 \pm 4}{2}$
• Step 7: Solve for the two potential values of $x$:
  $x_1 = \frac{12 + 4}{2} = 8$
  $x_2 = \frac{12 - 4}{2} = 4$

Thus, the solutions to the equation are $x_1 = 8$ and $x_2 = 4$.

Therefore, the correct answer is $x_1 = 8, x_2 = 4$, which corresponds to choice 1.