🧮 Quadratic Equation Solver

The quadratic formula, and what the discriminant decides

Roots come from the standard formula. The part under the square root is checked first, because its sign settles how many real answers exist:

D = b² − 4ac
x = (−b ± √D) / 2a           vertex x = −b / 2a

D > 0  two real roots     D = 0  one double root
D < 0  a conjugate pair, −b/2a ± (√−D / 2a) i

The defaults a = 1, b = −3, c = 2 give D = 9 − 8 = 1, so the roots are (3 ± 1) / 2: 2 and 1. The vertex sits at x = 1.5, which substituted back gives y = −0.25.

Reading the rest of the card

• Roots are where the curve meets the x-axis. A negative discriminant means it never does, so you get complex roots, not an error.
• The sign of a sets the shape. Positive opens upward, so the vertex is the minimum; negative opens downward, so it is the maximum.
• The axis of symmetry is the vertex's x value, the midpoint of the two roots.
• Quick sanity check: the roots should add to −b/a and multiply to c/a. Here 1 + 2 = 3 and 1 × 2 = 2, which matches.

Frequently asked questions