Questions: Solve the equation. [ (x-8)^2-4(x-8)-5=0 ] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Simplify your answer. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Express complex numbers in terms of i.) R. The solution set is the empty set.

Solution Steps

To solve the given equation, we can perform a substitution to simplify it. Let \( y = x - 8 \). This transforms the equation into a standard quadratic form: \( y^2 - 4y - 5 = 0 \). We can then solve this quadratic equation using the quadratic formula. Once we find the values of \( y \), we can substitute back to find the corresponding values of \( x \).

Let \( y = x - 8 \). The original equation transforms to: \[ y^2 - 4y - 5 = 0 \]

Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -4, c = -5 \): \[ y = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-5)}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 + 20}}{2} = \frac{4 \pm \sqrt{36}}{2} = \frac{4 \pm 6}{2} \] This gives us two solutions for \( y \): \[ y_1 = \frac{10}{2} = 5, \quad y_2 = \frac{-2}{2} = -1 \]

Now, substituting back to find \( x \): \[ x_1 = y_1 + 8 = 5 + 8 = 13 \] \[ x_2 = y_2 + 8 = -1 + 8 = 7 \]

Final Answer