1. **State the problem:** Solve the quadratic equation $-66 = -17x + x^2$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 - 17x + 66 = 0$$ 3. **Identify coefficients:** Here, $a = 1$, $b = -17$, and $c = 66$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$b^2 - 4ac = (-17)^2 - 4 \times 1 \times 66 = 289 - 264 = 25$$ 6. **Find the square root of the discriminant:** $$\sqrt{25} = 5$$ 7. **Substitute values into the formula:** $$x = \frac{-(-17) \pm 5}{2 \times 1} = \frac{17 \pm 5}{2}$$ 8. **Calculate the two solutions:** - For the plus sign: $$x = \frac{17 + 5}{2} = \frac{22}{2} = 11$$ - For the minus sign: $$x = \frac{17 - 5}{2} = \frac{12}{2} = 6$$ **Final answer:** The solutions are $x = 11$ and $x = 6$.