Question

solve the equation.

\\(x - 8)^2 - 2(x - 8) - 35 = 0\\

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the solution set is \\{\quad\\}.
(simplify your answer. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)

b. the solution set is the empty set.

Explanation:

🆕 New Concept Discovered: Solving Equations Quadratic in Form
Using a temporary variable to simplify complex expressions.

Step 1: Substitute a new variable

Let \( u = x - 8 \).

The original equation:

$$ (x - 8)^2 - 2(x - 8) - 35 = 0 $$

becomes:

$$ u^2 - 2u - 35 = 0 $$

Step 2: Solve the quadratic equation for \( u \)

We look for two numbers that multiply to \( -35 \) and add to \( -2 \). These numbers are \( -7 \) and \( 5 \).

$$ (u - 7)(u + 5) = 0 $$

This gives two possible values for \( u \):

$$ u = 7 \quad \text{or} \quad u = -5 $$

Step 3: Substitute back to solve for \( x \)

Now, replace \( u \) with \( x - 8 \) to find the values of \( x \).

For \( u = 7 \):

$$ x - 8 = 7 $$

$$ x = 15 $$

For \( u = -5 \):

$$ x - 8 = -5 $$

$$ x = 3 $$

Answer:

A. The solution set is \( \{3, 15\} \).