Short answer: x² − 3x + 2 = 0 has roots x = 1 and x = 2, because the discriminant is 1 and (−(−3) ± √1) ÷ 2 gives 2 and 1. The vertex sits at x = 1.5, y = −0.25.
Reading the discriminant first
b² − 4ac decides everything about the shape: positive means two crossings, zero means the parabola touches the axis once, negative means it never does and the roots are complex conjugates. Students often compute roots without checking it, then wonder why the answer looks wrong.
Worked example
2x² + 4x − 6 = 0: discriminant 16 + 48 = 64, so roots are (−4 ± 8) ÷ 4 = 1 and −3. The vertex is at x = −1, y = −8, and because a is positive the parabola opens upward with a minimum there.