Quadratic Equation Calculator
Solve ax² + bx + c = 0 with the quadratic formula: real or complex roots, discriminant, vertex and factored form, with working.
About this tool
Enter the coefficients a, b and c of a quadratic equation in standard form, ax² + bx + c = 0. The calculator finds the discriminant, tells you whether there are two real solutions, one double root or no real solutions (and gives the complex roots), and shows the working with the quadratic formula. You also get the vertex of the parabola, whether it opens up or down, the y-intercept and the factored form.
Frequently asked questions
For ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) ÷ 2a. For x² − 5x + 6 = 0, a = 1, b = −5 and c = 6, so x = (5 ± √(25 − 24)) ÷ 2 = (5 ± 1) ÷ 2, which gives x = 2 and x = 3.
The discriminant D = b² − 4ac is the part under the square root. If D > 0 there are two real solutions, if D = 0 there is exactly one (a double root), and if D < 0 there are no real solutions – only two complex ones. x² + 2x + 5 = 0 has D = 4 − 20 = −16, so its solutions are x = −1 ± 2i and its graph never touches the x-axis.
The vertex lies at x = −b ÷ 2a, and you get the y-value by putting that x back into the equation (or with y = c − b² ÷ 4a). For y = x² − 5x + 6 the vertex is x = 5 ÷ 2 = 2.5 and y = 6.25 − 12.5 + 6 = −0.25. If a is positive the vertex is the lowest point; if a is negative it is the highest.
Move every term to one side first. 2x² = 3 − 5x becomes 2x² + 5x − 3 = 0, so a = 2, b = 5 and c = −3. Then D = 25 + 24 = 49 and x = (−5 ± 7) ÷ 4, so x = 0.5 or x = −3. If a is 0 the equation is linear (bx + c = 0) and the solution is simply x = −c ÷ b.
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