Quadratic Formula Calculator

Everything about the roots before you compute them: b² − 4ac positive means two distinct real roots (the parabola crosses the x-axis twice), zero means one repeated root (it touches at the vertex), negative means a complex conjugate pair (it never crosses). On multiple-choice tests, computing just the discriminant often eliminates most answers in ten seconds.

When the factors are visible in a few seconds — x² − 5x + 6 begs to be (x−2)(x−3). When they aren't, the formula always works and is usually faster than hunting. This calculator shows the factored form whenever the roots come out as integers, which doubles as a check on your own factoring.

Nothing wrong — a negative discriminant means the parabola never touches the x-axis, and the roots involve i = √(−1), appearing as a conjugate pair like 2 ± 3i. Whether that's an acceptable answer depends on the course: algebra classes before complex numbers may phrase it as 'no real solutions,' which is the same fact.

Real roots are exactly where the parabola crosses the x-axis, and the vertex (shown in the steps as −b/2a for x) is its turning point, midway between the roots. Plot the same equation in the graphing calculator to see it — connecting the algebra to the picture is genuinely the fastest way to make quadratics stop feeling arbitrary.