Solve ax²+bx+c=0 and see roots, discriminant, vertex and SVG parabola
A quadratic solver finds the solutions of a quadratic equation in the form ax² + bx + c = 0, given the three coefficients a, b and c. It returns the roots — the values of x where the equation equals zero — along with the discriminant, the vertex of the parabola and a plotted curve, so you see not just the answer but the shape behind it. Quadratics appear throughout maths, physics and engineering, from projectile motion to optimisation, and solving them by hand with the quadratic formula is exactly the kind of careful arithmetic that is easy to slip on.
The key to a quadratic is the discriminant, the b² − 4ac part of the formula, because its sign tells you the nature of the roots before you finish solving: positive means two distinct real roots, zero means one repeated root, and negative means two complex roots with no real x-intercepts. Seeing the discriminant, the vertex (the parabola's turning point) and the graph together turns the abstract formula into an intuitive picture. This tool computes everything in your browser and draws the parabola so the solution is visual as well as numeric.
The discriminant is b² − 4ac, the part under the square root in the quadratic formula. Its sign reveals the roots before you finish: positive gives two distinct real roots, zero gives one repeated root, and negative gives two complex roots with no real solutions — so the graph does not cross the x-axis.
The vertex is the parabola's turning point — its lowest point if it opens upward, or highest if it opens downward. It marks where the curve changes direction and is the maximum or minimum value of the quadratic, which matters in optimisation problems.
When the discriminant is negative, the parabola never touches the x-axis, so there are no real values of x that make the equation zero. The roots are complex numbers instead. A graph makes this clear: the curve sits entirely above or below the axis.
If a is zero the equation is no longer quadratic — it becomes linear (bx + c = 0), which has a single straightforward solution rather than the parabola and up-to-two roots of a true quadratic. A quadratic requires a non-zero a coefficient.
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