We start with the given equation:
[tex]\[ (x - 12)(x + 4) = 9 \][/tex]
First, we need to expand the left-hand side using the distributive property:
[tex]\[ (x - 12)(x + 4) = x^2 + 4x - 12x - 48 \][/tex]
Simplify this:
[tex]\[ x^2 + 4x - 12x - 48 = x^2 - 8x - 48 \][/tex]
So, the equation becomes:
[tex]\[ x^2 - 8x - 48 = 9 \][/tex]
Next, we move the constant on the right-hand side to the left-hand side by subtracting 9 from both sides:
[tex]\[ x^2 - 8x - 48 - 9 = 0 \][/tex]
Simplify to:
[tex]\[ x^2 - 8x - 57 = 0 \][/tex]
To solve this using the method of completing the square, we first need to isolate the x-terms:
[tex]\[ x^2 - 8x = 57 \][/tex]
Next, we complete the square by adding and subtracting the square of half the coefficient of [tex]\(x\)[/tex]:
Take half of the coefficient of [tex]\(x\)[/tex] (which is -8), square it, and add it to both sides:
[tex]\[ \left( \frac{-8}{2} \right)^2 = 16 \][/tex]
Add 16 to both sides:
[tex]\[ x^2 - 8x + 16 = 57 + 16 \][/tex]
This can be written as:
[tex]\[ (x - 4)^2 = 73 \][/tex]
To solve for [tex]\(x\)[/tex], we take the square root of both sides:
[tex]\[ x - 4 = \pm \sqrt{73} \][/tex]
Therefore, solving for [tex]\(x\)[/tex]:
[tex]\[ x = 4 \pm \sqrt{73} \][/tex]
So the solutions are:
[tex]\[ x = 4 + \sqrt{73} \][/tex]
[tex]\[ x = 4 - \sqrt{73} \][/tex]
These results indicate that the correct answer is:
[tex]\[ x = 4 \pm \sqrt{73} \][/tex]
