Quadratic Formula Calculator
Enter your values below and click Calculate for instant results.
ax² + bx + c = 0
Solve any quadratic equation ax² + bx + c = 0. Shows all roots, discriminant, vertex, and root type.
Enter your values below and click Calculate for instant results.
The quadratic formula solves any equation of the form ax squared plus bx plus c equals zero. The formula is: x equals negative b, plus or minus the square root of (b squared minus 4ac), all divided by 2a. It always works, regardless of whether the coefficients are integers, decimals, positive, or negative.
The discriminant (b squared minus 4ac) tells you the nature of the solutions before you calculate them. If it is positive, there are two distinct real solutions. If it is zero, there is one repeated solution. If it is negative, the solutions are complex numbers (involving the square root of a negative number).
A positive discriminant means the parabola (graph of the quadratic) crosses the x-axis in two places. A zero discriminant means it touches the x-axis at exactly one point (the vertex). A negative discriminant means the parabola never crosses the x-axis, indicating no real solutions exist for the equation.
The vertex of the parabola is at x equals -b divided by 2a. This is the turning point of the curve: the maximum point if the parabola opens downward (a is negative) or the minimum point if it opens upward (a is positive). This calculator provides the vertex coordinates along with the roots.