5.3.2, the value of the expression : (2) \( -x^{2}+3 x-2 \) \( = \)

To evaluate the expression \(-x^{2}+3x-2\) at \(x = 2\), simply plug in the value of \(x\): \[ -x^{2}+3x-2 = -2^{2} + 3(2) - 2 = -4 + 6 - 2 = 0. \] So, the value of the expression is \(0\). Now, let’s think about how quadratic equations like this one show up in real life — from designing the perfect parabolic arches for bridges to modeling the trajectory of a basketball! Understanding the behavior of quadratic equations can help us predict outcomes in various fields like sports, engineering, and even economics. And here’s a little tip: when dealing with quadratic expressions, remember to check the signs of the coefficients. Like in our equation, if the leading coefficient (the one in front of \(x^2\)) is negative, the parabola opens downwards, which can affect where the maximum point is located. Always visualize the graph if you can; it makes spotting these features much easier!