Looking for a quadratic equation solver with steps formula? This free tool gives you both roots, discriminant and vertex of any quadratic equation. Just enter your numbers and press Calculate — results are instant.
A quadratic equation has the form ax² + bx + c = 0, and its roots come from the quadratic formula: x equals minus b, plus or minus the square root of the discriminant, all over 2a. The discriminant (b² − 4ac) tells you what kind of roots to expect: positive means two real roots, zero means one repeated root, and negative means two complex roots. The calculator also gives the vertex, the turning point of the parabola.
Example: Solve x² − 5x + 6 = 0 (a=1, b=−5, c=6)
Root 1 (x₁): 3.000000
The discriminant (b² − 4ac) tells you what kind of roots to expect before you solve: positive means two different real roots, zero means one repeated root, and negative means two complex roots.
The equation has no real roots — its graph never touches the x-axis. The calculator then gives the two complex roots, in the form a ± bi.
Factor it. For x² − 5x + 6 = 0, the roots 2 and 3 give the factors (x − 2)(x − 3), which multiplies back to x² − 5x + 6.