The equation provided is:

$(x - 4)^2 + 3(x - 4) - 18 = 0$

Let's solve this step-by-step.

Step 1: Substitute $y = x - 4$

This substitution simplifies the equation. Let $y = x - 4$, so the equation becomes:

$y^2 + 3y - 18 = 0$

Step 2: Solve the Quadratic Equation $y^2 + 3y - 18 = 0$

We need to find the values of $y$ that satisfy this equation. This is a standard quadratic equation in the form $ay^2 + by + c = 0$, where:

• $a = 1$
• $b = 3$
• $c = -18$

Using the quadratic formula:

$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute $a = 1$, $b = 3$, and $c = -18$:

$y = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-18)}}{2 \cdot 1}$

$y = \frac{-3 \pm \sqrt{9 + 72}}{2}$

$y = \frac{-3 \pm \sqrt{81}}{2}$

$y = \frac{-3 \pm 9}{2}$

This gives us two solutions for $y$:

1. $y = \frac{-3 + 9}{2} = 3$
2. $y = \frac{-3 - 9}{2} = -6$

Step 3: Substitute Back for $x$

Now that we know $y = 3$ and $y = -6$, we substitute back $y = x - 4$:

1. For $y = 3$: $x - 4 = 3 \Rightarrow x = 7$

2. For $y = -6$: $x - 4 = -6 \Rightarrow x = -2$

Solution

The real solutions for $x$ are $x = 7$ and $x = -2$. Therefore, the correct answer is:

$\{ -2, 7 \}$

Based on the options, the correct choice is:

• Option A: { -2, 7 }

Would you like further details on any part of the solution?

Here are some related questions to deepen your understanding:

1. How can you verify the solutions of a quadratic equation by substitution?
2. What does it mean to complete the square in a quadratic equation?
3. How would the solution change if the constant term were positive instead of negative?
4. What are some practical applications of solving quadratic equations?
5. How can substitution simplify complex algebraic equations?

Tip: Always consider using substitution for simplifying complex expressions, especially when terms repeat in a structured way.