To solve the equation [tex]\((x-12)(x+4) = 9\)[/tex] using completing the square, follow these steps:
1. Expand the equation:
Start by expanding the left-hand side of the equation:
[tex]\[(x-12)(x+4) = x^2 + 4x - 12x - 48 = x^2 - 8x - 48\][/tex]
Therefore, the equation is:
[tex]\[x^2 - 8x - 48 = 9\][/tex]
2. Move all terms to one side:
To set the equation to 0, subtract 9 from both sides:
[tex]\[x^2 - 8x - 48 - 9 = 0\][/tex]
Simplify it to:
[tex]\[x^2 - 8x - 57 = 0\][/tex]
3. Complete the square:
Add and subtract the square of half the coefficient of [tex]\(x\)[/tex]. Here, the coefficient of [tex]\(x\)[/tex] is -8, so half of it is -4, and its square is 16. Therefore:
[tex]\[x^2 - 8x - 57 = (x^2 - 8x + 16) - 16 - 57\][/tex]
Simplify inside the parentheses to complete the square:
[tex]\[(x-4)^2 - 73 = 0\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Isolate the squared term:
[tex]\[(x-4)^2 = 73\][/tex]
Take the square root of both sides:
[tex]\[x-4 = \pm \sqrt{73}\][/tex]
Finally, solve for [tex]\(x\)[/tex]:
[tex]\[x = 4 \pm \sqrt{73}\][/tex]
Therefore, the solutions to the equation [tex]\((x-12)(x+4) = 9\)[/tex] are:
[tex]\[x = 4 + \sqrt{73} \quad \text{or} \quad x = 4 - \sqrt{73}\][/tex]
