(A82) Solve a quadratic equation by completing the square Fill in each blank with a whole number. \( x^{2}-6 x+40=(x-3 \)

To solve the quadratic equation \( x^{2}-6x+40 \) by completing the square, first, we rearrange it as \( x^{2}-6x = -40 \). Next, we take half of the linear coefficient (-6), square it (which gives us 9), and then add it to both sides: \[ x^{2}-6x + 9 = -40 + 9 \] This simplifies to: \[ (x-3)^2 = -31 \] Now, to find the solutions, we can write: \[ x-3 = \pm i\sqrt{31} \] So, we ultimately have: \[ x = 3 \pm i\sqrt{31} \] This shows that our solutions are complex numbers, which is a fascinating outcome when working with quadratic equations! Now for a twist – did you know that the technique of completing the square dates back to ancient Babylonian mathematics? They were solving quadratic-like problems as far back as 2000 BCE! This method not only showcases the evolution of math but also highlights the brilliance of mathematicians from centuries ago who laid the foundation for modern algebra. Completing the square is not just an academic exercise; it plays a key role in various fields today! Engineers use this method when designing parabolic structures, like bridges and arches, to find optimal shapes that can bear weight effectively. It also pops up in computer graphics for rendering curves and animations, proving that algebra isn't just for classrooms; it's at the heart of innovation in our everyday lives!