How this calculator works
Enter the three coefficients of a quadratic equation in standard form, ax² + bx + c = 0, and this calculator returns both roots using the quadratic formula, along with the discriminant and the vertex of the parabola y = ax² + bx + c.
The discriminant, b² − 4ac, tells you what kind of roots to expect before you even compute them: positive means two distinct real roots, zero means one repeated real root (the parabola just touches the x-axis), and negative means two complex roots that come as a conjugate pair. If you set a to 0, the x² term disappears and the calculator solves the remaining linear equation bx + c = 0 instead.
How do you solve a quadratic equation?
Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ (2a)Discriminant: D = b² − 4acVertex: x = −b ÷ (2a), y = a·x² + b·x + c at that xWhen D < 0, the square root of a negative number is handled as an imaginary number: √(−D) = i·√D, giving complex roots of the form p ± qi.
Worked example: x² − 3x + 2 = 0
- Identify coefficients: a = 1, b = −3, c = 2.
- Discriminant: D = (−3)² − 4(1)(2) = 9 − 8 = 1, which is positive, so expect two real roots.
- Apply the formula: x = (3 ± √1) ÷ 2 = (3 ± 1) ÷ 2.
- The two roots are x = 4 ÷ 2 = 2 and x = 2 ÷ 2 = 1, matching the factored form (x − 1)(x − 2) = 0.
Frequently asked questions
What does a negative discriminant mean?
It means the parabola never crosses the x-axis, so there are no real roots. The two solutions are a complex conjugate pair, written as p ± qi, where p is the real part and q the imaginary part.
What happens if a = 0?
The equation is no longer quadratic. This calculator falls back to solving the linear equation bx + c = 0, giving the single root x = −c ÷ b, and notes that it did so.
What is the vertex used for?
The vertex (−b ÷ 2a, f(−b ÷ 2a)) is the parabola's minimum point when a > 0 or maximum point when a < 0. It is also the axis of symmetry: the two roots (when real) sit equally spaced on either side of the vertex's x-coordinate.
Can I check the roots myself?
Yes. Substitute each root back into ax² + bx + c and the result should be 0 (or extremely close to it after rounding). This calculator's own test suite verifies every golden example the same way.
Why do some quadratics have a double root?
A double (repeated) root happens exactly when the discriminant is 0, which means the expression is a perfect square, like x² + 2x + 1 = (x + 1)². The parabola touches the x-axis at one point instead of crossing it twice.