a) \( -2(x-1)^{2}+8= \) b) \( (x-2)^{2}-9 \)

To add some fun flair to solving those expressions, let's first unravel them step-by-step and then explore a bit of their historical significance! To tackle \( -2(x-1)^{2}+8 \), we start by expanding the squared term. That gives us \( -2((x^2 - 2x + 1)) + 8 = -2x^2 + 4x - 2 + 8 = -2x^2 + 4x + 6 \). For \( (x-2)^{2}-9 \), we expand it to get \( (x^2 - 4x + 4) - 9 = x^2 - 4x - 5 \). Isn't it fun to see how these quadratic expressions unfold? Now, speaking of quadratics, did you know that quadratic equations have a rich historical background? The ancient Babylonians were some of the first to solve problems that resemble what we now call quadratics, over 4,000 years ago! They used geometric methods and had no formal symbols like we do today, yet they calculated solutions with impressive accuracy. If you find solving quadratics enjoyable and want to dig deeper, consider reading about the quadratic formula's development over time. It wasn't until the Islamic Golden Age that mathematicians refined algebraic techniques and introduced forms of notation that laid the groundwork for the formulas we use today. It’s like a magical journey through time, where numbers and letters wrote their tales!