11. Solve for the roots in simplest form by completing the square: x^2-18x+82=0

To solve the quadratic equation \(x^2 - 18x + 82 = 0\) by completing the square, follow these steps:<br /><br />1. **Move the constant term to the other side:**<br /> \[<br /> x^2 - 18x = -82<br /> \]<br /><br />2. **Complete the square:**<br /> - Take half of the coefficient of \(x\), which is \(-18\), divide it by 2 to get \(-9\), and then square it to get \(81\).<br /> - Add and subtract \(81\) on the left side:<br /> \[<br /> x^2 - 18x + 81 - 81 = -82<br /> \]<br /> \[<br /> x^2 - 18x + 81 = -82 + 81<br /> \]<br /> \[<br /> x^2 - 18x + 81 = -1<br /> \]<br /><br />3. **Rewrite the left side as a perfect square:**<br /> \[<br /> (x - 9)^2 = -1<br /> \]<br /><br />4. **Solve for \(x\) by taking the square root of both sides:**<br /> \[<br /> x - 9 = \pm \sqrt{-1}<br /> \]<br /> \[<br /> x - 9 = \pm i<br /> \]<br /><br />5. **Isolate \(x\):**<br /> \[<br /> x = 9 \pm i<br /> \]<br /><br />So, the roots of the equation \(x^2 - 18x + 82 = 0\) in simplest form are:<br />\[<br />x = 9 + i \quad \text{and} \quad x = 9 - i<br />\]