Quadratic Equation Explorer

For any equation ax² + bx + c = 0 with a not 0, the quadratic formula x = (-b ± √(b² - 4ac)) / 2a gives both solutions at once. It works even when factoring is ugly or impossible. For example, with a = 1, b = -2, c = -3 the formula gives (2 ± 4) / 2, so x = 3 and x = -1. The explorer runs this live: change any coefficient and the marked roots slide along the x-axis.

The discriminant is the part under the square root, Δ = b² - 4ac, and its sign alone answers "how many real roots?" before you solve anything. Positive Δ means two real roots, Δ = 0 means one repeated root where the vertex touches the x-axis, and negative Δ means no real roots. Watch the Δ readout as you raise c: the moment it crosses zero the parabola lifts off the axis.

The x-coordinate of the vertex is always h = -b / 2a, and substituting h back in gives the y-coordinate k. The vertical line x = h is the axis of symmetry: the parabola is a mirror image across it. For example, x² - 2x - 3 has h = 1 and k = -4, so the vertex is (1, -4). In the explorer the dashed line and the vertex dot track this for you continuously.

The sign of a sets the direction the parabola opens: positive a opens upward with the vertex as a minimum, negative a opens downward with the vertex as a maximum. At exactly a = 0 the x² term disappears and the equation is not quadratic any more, just the line y = bx + c. Slide a through zero and watch the parabola flatten into a line and flip over.

Yes. The formula panel doubles as a quadratic equation solver: type your own a, b, and c into the number boxes and the parabola, roots, vertex, and discriminant all follow. The panel also shows the discriminant worked with your numbers, which is handy for checking each step of homework against the picture.

It is the vertical line x = -b / 2a that cuts the parabola into two mirror halves, passing exactly through the vertex. Everything on the curve pairs up across it, which is why the two roots always sit at equal distances from it. For example, x² - 2x - 3 has axis x = 1, and its roots -1 and 3 are each two units away. The explorer draws the axis as a dashed line so the mirror symmetry is visible while you drag.

Compute h = -b / 2a and k by substituting h back in, then write y = a(x - h)² + k. For example, x² - 2x - 3 gives h = 1 and k = -4, so the vertex form is y = (x - 1)² - 4. This is the same result completing the square produces, just reached through the vertex coordinates. Drag the vertex in the explorer and it performs the reverse conversion live, solving b and c from your chosen h and k.

Factor when the roots are small whole numbers you can spot quickly: x² - 2x - 3 factors as (x - 3)(x + 1) in one step. Reach for the quadratic formula when factoring is not obvious within a few seconds, because it works on every quadratic, including ones with fractional, irrational, or no real roots. A useful habit: check the discriminant first, since a perfect-square Δ (like 16) signals that a clean factoring exists.

No. Everything runs in your browser on the HTML canvas: coefficients, roots, and the drawn parabola never leave your device. There is no account and the page keeps working offline once loaded.