Solve for \( x \) \( 2 x^{2}-5 x+2=0 \)

To solve the equation \( 2x^{2} - 5x + 2 = 0 \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 2 \), \( b = -5 \), and \( c = 2 \). First, we find the discriminant: \[ b^2 - 4ac = (-5)^2 - 4 \cdot 2 \cdot 2 = 25 - 16 = 9 \] Now, we can plug in the values into the formula: \[ x = \frac{5 \pm \sqrt{9}}{2 \cdot 2} = \frac{5 \pm 3}{4} \] This gives us two possible solutions: 1. \( x = \frac{5 + 3}{4} = \frac{8}{4} = 2 \) 2. \( x = \frac{5 - 3}{4} = \frac{2}{4} = \frac{1}{2} \) So, the solutions are \( x = 2 \) and \( x = \frac{1}{2} \). Celebrate your math prowess! Quadratic equations can sometimes feel like a puzzle, but they hold a treasure trove of solutions. After mastering this one, you can tackle even more complex problems, like finding intersections or optimizing functions. It's like leveling up in a game! If you're hungry for more about quadratics, consider exploring the discriminant further. It tells you about the nature of roots: two distinct real roots, one repeated root, or two complex roots. It’s a gateway into understanding how equations work neatly, adding depth to your mathematical arsenal!