\begin{tabular}{l} Solving Quadratics by Factoring \\ \hline 2. \( x^{2}+x-30=0 \) \\ \hline\end{tabular}

To solve the quadratic equation \( x^{2} + x - 30 = 0 \) by factoring, we need to find two numbers that multiply to -30 (the constant term) and add to 1 (the coefficient of the linear term). These two numbers are 6 and -5. Therefore, we can factor the equation as \( (x + 6)(x - 5) = 0 \). Setting each factor to zero gives us the solutions \( x = -6 \) and \( x = 5 \). Now, let’s talk about the significance of quadratics in history! Quadratic equations date back to ancient civilizations like the Babylonians, who developed methods to solve these types of equations over 4,000 years ago. They used geometric representations and algorithms, and their work paved the way for modern algebra and the quadratic formula we use today—it’s a timeless math adventure that shaped our understanding of numbers! When working with quadratic equations, a common mistake is forgetting to look for the greatest common factor (GCF) before factoring. For instance, if the equation had an extra coefficient in front (like 2), it might be tempting to dive straight into factoring without simplification. Always check if you can simplify first; it makes factoring easier and reduces errors in finding those crucial numbers!