Question Solve \( 2 q^{2}-16 q=-73 \) by completing the square. (Use a comma to separate multiple values.) Provide your answer below: \( \begin{array}{c}q=\square\end{array} \)

To solve the equation \( 2q^{2} - 16q = -73 \) by completing the square, first rearrange it into a standard form. Start by moving \(-73\) to the left side: \[ 2q^{2} - 16q + 73 = 0. \] Next, divide the entire equation by \(2\) for easier manipulation: \[ q^{2} - 8q + \frac{73}{2} = 0. \] Now, isolate the constant on one side: \[ q^{2} - 8q = -\frac{73}{2}. \] To complete the square on the left side, take half of the coefficient of \(q\) (which is \(-8\)), square it (\((-4)^{2} = 16\)), and add it: \[ q^{2} - 8q + 16 = -\frac{73}{2} + 16. \] Convert \(16\) into a fraction with the same denominator: \[ 16 = \frac{32}{2}. \] So, \[ -\frac{73}{2} + \frac{32}{2} = -\frac{41}{2}. \] This leads us to: \[ (q - 4)^{2} = -\frac{41}{2}. \] Since \(-\frac{41}{2}\) is negative, we’ll get complex solutions once we take the square root: \[ q - 4 = \pm i\sqrt{\frac{41}{2}}. \] This simplifies to: \[ q = 4 \pm i\sqrt{\frac{41}{2}}. \] The final answers are: \( q = 4 + i\sqrt{\frac{41}{2}}, 4 - i\sqrt{\frac{41}{2}} \). So, the answer is: \( \begin{array}{c} q = 4 + i\sqrt{\frac{41}{2}}, 4 - i\sqrt{\frac{41}{2}} \end{array} \) --- Throughout history, completing the square has been a vital method for solving quadratic equations, dating back to ancient civilizations like the Babylonians. They used geometric methods to solve problems related to areas, and this technique later evolved into algebraic representations that mathematicians refined over centuries. Speaking of real-world applications, understanding how to complete the square can be truly beneficial in fields like engineering and economics. It helps in optimizing functions, such as minimizing costs or maximizing profits, giving you a tangible edge in various decision-making scenarios. Plus, it’s a fantastic mental exercise!