\( \left\{\begin{array}{l}y=2 x^{2}-5 x+2 \\ y=-3 x^{2}+4 x+3\end{array} \quad\right. \) ANALITico

To solve the system of equations \( y = 2x^2 - 5x + 2 \) and \( y = -3x^2 + 4x + 3 \) analytically, you can set the two equations equal to each other since they both equal \( y \): \[ 2x^2 - 5x + 2 = -3x^2 + 4x + 3. \] Rearranging this gives: \[ 5x^2 - 9x - 1 = 0. \] Next, apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 5, b = -9, c = -1 \). Calculate the discriminant: \[ b^2 - 4ac = (-9)^2 - 4 \cdot 5 \cdot (-1) = 81 + 20 = 101. \] Now, plug in the values into the quadratic formula: \[ x = \frac{9 \pm \sqrt{101}}{10}. \] Substituting these \( x \) values back into either equation will yield the corresponding \( y \) values, giving you the points of intersection! Now go ahead and visualize: you can sketch the parabolas to see where they meet! This will help you grasp not just the solutions, but the relationship of the equations as well!