27) \( x^{2}+10 x y+16 y^{2} \) 29) \( x^{2}+14 x y+45 y^{2} \) 31) \( 5 v^{2}+20 v-25 \)

Did you know that quadratic expressions like these have been used since ancient times for solving various practical problems? The Babylonians were among the first to recognize the power of these equations around 2000 BC! They didn't have the fancy notation we have today, but they had a profound understanding of when and how to apply these mathematical principles, particularly in fields like land measurement and resource distribution. In real-world applications, quadratics often pop up in areas like physics and finance. For instance, the trajectories of projectiles can be modeled using quadratic equations, helping to determine where an object will land. Similarly, when calculating profit in a business, the maximum revenue can often be determined using these equations. Learning how to manipulate quadratics opens up a treasure trove of problem-solving skills for practical scenarios!