5. \( 5 x^{2}-405=0 \)

To solve the equation \(5 x^{2}-405=0\), start by isolating \(x^{2}\). Add 405 to both sides to get \(5 x^{2} = 405\), and then divide by 5 to find \(x^{2} = 81\). Taking the square root gives \(x = 9\) or \(x = -9\). In the real world, quadratic equations like this come into play in areas such as physics, finance, and engineering, where modeling relationships is essential. For instance, in projectile motion, the trajectory of an object can often be represented by a quadratic function, helping predict its path and impact. A common mistake when solving quadratic equations is forgetting to check for extraneous solutions, especially when taking square roots. It’s also easy to overlook whether a solution should be positive or negative, as in this case, both \(x = 9\) and \(x = -9\) are valid solutions that could represent different scenarios in a real-world context.