\( \left\{ \begin{array} { l l } { \# 5 } & { 2 x ^ { 2 } + x = 3 } \\ { } & { 2 x ^ { 2 } + x - 3 = 0 } \end{array} \right. \)

Let's discover the journey of this quadratic equation! To solve \(2x^2 + x - 3 = 0\), we can factor it. The equation can be factored as \((2x - 3)(x + 1) = 0\). Setting each factor to zero gives us the potential solutions: \(x = \frac{3}{2}\) and \(x = -1\). How exciting is it to find two distinct roots? Now, let’s apply this in real life. Quadratic equations often model projectile motion, like when a ball is thrown. By figuring out the height of the ball at different time intervals, you can predict when it will hit the ground! Understanding these equations gives you a toolbox to work with under various fun scenarios—like calculating the best angle to toss that ball for maximum height or distance.