Quadratic Equation Solver
Solve ax² + bx + c = 0 for real or complex roots, with the discriminant shown.
Comment ça marche
A quadratic equation has the form ax² + bx + c = 0, where a, b, and c are numbers you provide (with a not equal to zero). This solver finds the value or values of x that make the equation true, called the roots or solutions.
It works by first computing the discriminant, Δ = b² − 4ac, then applying the quadratic formula: x = (−b ± √Δ) / 2a. The discriminant alone tells you what kind of roots to expect before even finishing the calculation.
If the discriminant is positive, there are two distinct real roots — for x² − 5x + 6 = 0 (a=1, b=−5, c=6), Δ = 25 − 24 = 1, giving roots x = 3 and x = 2. If the discriminant is exactly zero, there's exactly one repeated real root, since ±√0 collapses to a single value. If the discriminant is negative, there are no real roots at all — instead, two complex conjugate roots involving i (the imaginary unit, √−1), shown in the form p ± qi.
Quadratic equations show up constantly in physics and engineering — the height of a thrown object over time follows a quadratic (parabolic) path, and finding when it hits the ground means solving a quadratic equation for time. They also appear in optimization problems, like finding the dimensions that maximize area for a fixed amount of fencing.
For equations that factor neatly (whole-number roots), factoring by inspection is often faster than the full formula — but the quadratic formula always works, even when factoring isn't obvious or the roots are irrational or complex, which is why it's the reliable fallback method.
The sign of a also tells you the shape of the parabola this equation describes: a positive a means the parabola opens upward (like a U), while a negative a means it opens downward — useful context if you're graphing the equation, not just solving it.
What if the discriminant is negative?
The equation has two complex conjugate roots, shown in the form a ± bi.
What does the discriminant tell me before I even solve the equation?
Its sign predicts the type of roots: positive means two real roots, zero means one repeated real root, and negative means two complex roots — no need to finish the calculation to know which case applies.
Why can't 'a' be zero?
If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic — it would need a different, simpler solving method entirely.
What's a real-world example of a quadratic equation?
The height of a ball thrown into the air over time follows a quadratic path due to gravity — solving for when height equals zero tells you when it lands.
Is factoring faster than the quadratic formula?
For equations with clean, whole-number roots, yes — but factoring isn't always obvious or possible, while the quadratic formula reliably works for every quadratic equation, including ones with irrational or complex roots.
What does it mean for a parabola to 'open upward' or 'downward'?
It describes the shape's orientation: a positive leading coefficient (a) makes a U-shape opening upward, while a negative a makes an upside-down U opening downward.
Can a quadratic equation have exactly one solution?
Yes — when the discriminant equals exactly zero, the two roots from the ± in the formula collapse into the same single value, called a repeated or double root.
What is the imaginary unit i?
i is defined as the square root of −1, used to express roots of negative numbers — complex roots from a negative discriminant are written using i, in the form p ± qi.
How do I verify a root is correct?
Substitute it back into the original equation ax² + bx + c — if the result equals zero, the root is correct.
What is completing the square, and how does it relate to the quadratic formula?
It's an alternative solving method that rewrites the equation as a perfect square plus a constant — in fact, the quadratic formula itself is derived by completing the square on the general equation ax² + bx + c = 0.
Can this solver handle equations not already in ax² + bx + c = 0 form?
Rearrange the equation to that standard form first (moving all terms to one side, set equal to zero), then enter the resulting a, b, and c coefficients.