Question

solve the equation for all values of x by completing the square. express your answer in simplest form.\\(x^2 - 18x = -68\\)

Explanation:

Step1: Complete the square for the x-terms

To complete the square for \(x^2 - 18x\), we take half of the coefficient of \(x\), square it, and add it to both sides. The coefficient of \(x\) is \(-18\), half of it is \(\frac{-18}{2}=-9\), and squaring it gives \((-9)^2 = 81\). So we add 81 to both sides of the equation:
\[
x^2 - 18x + 81=-68 + 81
\]

Step2: Rewrite the left side as a perfect square

The left side \(x^2 - 18x + 81\) can be written as \((x - 9)^2\), and the right side simplifies to \(13\):
\[
(x - 9)^2=13
\]

Step3: Solve for x

Take the square root of both sides:
\[
x - 9=\pm\sqrt{13}
\]
Then add 9 to both sides to solve for \(x\):
\[
x = 9\pm\sqrt{13}
\]

Answer:

\(x = 9 + \sqrt{13}\) or \(x = 9 - \sqrt{13}\) (or combined as \(x = 9\pm\sqrt{13}\))