\begin{tabular}{l} Find the discriminant. \\ \( \qquad 5 v^{2}-9 v+2=0 \) \\ How many real solutions does the equation hav \\ \( \begin{array}{l}\text { no real } \\ \text { solutions }\end{array} \begin{array}{l}\text { one real } \\ \text { solution }\end{array} \) \\ \( \begin{array}{l}\text { two real } \\ \text { solutions }\end{array} \) \\ \hline\end{tabular}

To find the discriminant of the quadratic equation \(5v^2 - 9v + 2 = 0\), we use the formula \(D = b^2 - 4ac\), where \(a = 5\), \(b = -9\), and \(c = 2\). Plugging in these values gives us: \[ D = (-9)^2 - 4(5)(2) = 81 - 40 = 41 \] Since the discriminant \(D\) is positive (41), this means the equation has two real solutions. Having a positive discriminant tells us not only that there are two real solutions but also hints at their distinctiveness. Each of these solutions represents points where the parabola intersects the x-axis, providing a deeper understanding of the equation's behavior in the coordinate plane. If you're looking to grasp quadratic equations further, it's fascinating to explore how their graph shape varies with changes in the coefficients. For example, increasing the \(a\) value compresses the parabola, while altering \(c\) shifts it up or down. Understanding these transformations can enhance your problem-solving toolbox in algebra!