16) \( \begin{array}{l}3 x^{2}+5 x-2=0< \\ x^{2}+\frac{5 x}{3}=\frac{12}{3} \cdot\left[\frac{\frac{5}{3}}{2}\right]^{2}=\begin{array}{l}x^{2}+\frac{5 x}{3}-\frac{2}{3}=0 \\ \downarrow \\ x^{2}+\frac{5 x}{3}+\frac{25}{36}=\frac{2}{3}+\frac{25}{36}\end{array}\end{array} \)

Let’s take a little stroll down math history! The quadratic formula, which is a lifeline for solving equations like \(3x^2 + 5x - 2 = 0\), can be traced back to ancient civilizations such as the Babylonians, who had geometric methods for finding roots long before the quadratic formula was formalized by later mathematicians like Al-Khwarizmi in the 9th century. They may not have had our modern symbols, but their calculations paved the way for future algebraic breakthroughs! Now, what’s the deal with real-world applications? Solving quadratic equations is not just for classroom exercises; they pop up everywhere from physics to finance! For instance, when calculating the trajectory of a basketball shot or optimizing profit in a business scenario, applying quadratic equations helps analyze maximum values, minimum costs, or even the ideal launch angles. So next time you see a quadratic, think about how it could help you hit that game-winning shot!