Question

solve the equation.

\\(x - 9)^2 - 2(x - 9) - 15 = 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 substitution to simplify complex equations.

Step 1: Substitute to simplify

Let \( u = x - 9 \).

The equation becomes:

$$ u^2 - 2u - 15 = 0 $$

Step 2: Factor the quadratic equation

Find two numbers that multiply to \(-15\) and add to \(-2\). These numbers are \(-5\) and \(3\):

$$ (u - 5)(u + 3) = 0 $$

Set each factor to zero:

$$ u = 5 \quad \text{or} \quad u = -3 $$

Step 3: Substitute back and solve for x

Replace \( u \) with \( x - 9 \):

For \( u = 5 \):

$$ x - 9 = 5 $$

$$ x = 14 $$

For \( u = -3 \):

$$ x - 9 = -3 $$

$$ x = 6 $$

Answer:

A. The solution set is \(\{6, 14\}\).