A quadratic equation has the form ax² + bx + c = 0, where a is not zero, and it graphs as a parabola. This quadratic calculator solves it for you in one step: you enter the three coefficients a, b, and c, and it computes the discriminant, the two roots (real or complex), the vertex of the parabola, and the direction it opens — showing you exactly how the parabola sits on the x-axis.
The same math appears far more often than you might expect: projectile motion (a ball thrown in the air), maximum profit models, bridge arches, satellite dishes, and area problems all reduce to a parabola. Whether you are checking homework, designing something, or debugging a physics simulation, entering three numbers beats reaching for the quadratic formula on paper every time.
How to Use
- Enter Coefficient a — the number in front of x². It cannot be zero; if a = 0 the equation is linear, not quadratic. Try changing the sign of a and watch the parabola flip from opening upward to opening downward.
- Enter Coefficient b — the number in front of x. Negative b values are common (for example, x² − 5x + 6 has b = −5), so type the minus sign as well.
- Enter the Constant c — the term with no x. This is the y-intercept of the parabola; c = 0 simply means the curve passes through the origin.
- Read the result card. The Discriminant (b² − 4ac) tells you immediately what kind of roots to expect: positive means two real roots, zero means a single repeated root, and negative means two complex roots with no crossing of the x-axis.
- Use Root 1 and Root 2 as the two solutions. When the discriminant is negative they are printed in the form real ± imaginary i, so you can read them straight off for complex-number work.
- Check the Vertex and Parabola Direction lines for the real-world reading: the vertex is the maximum (if it opens down) or minimum (if it opens up) point of the curve.
The Quadratic Formula Explained
For ax² + bx + c = 0, the two solutions come from the quadratic formula:
x = (−b ± √(b² − 4ac)) / (2a)
The part under the square root, D = b² − 4ac, is the discriminant, and it decides everything:
- D > 0 — the parabola crosses the x-axis at two distinct points: two real roots.
- D = 0 — the parabola just touches the x-axis at its vertex: one repeated (double) root.
- D < 0 — the parabola never reaches the x-axis: the roots are complex conjugates, written as real ± imaginary·i.
The vertex sits at x = −b/(2a), exactly halfway between the two real roots, and plugging that x back into the equation gives the vertex height y = c − b²/(4a). This is why, when a is positive, the vertex is the minimum value of the expression — and why problems like "what price maximizes profit?" always ask for that vertex.
Examples
Example 1 — The classic textbook case. Solve x² − 5x + 6 = 0: a = 1, b = −5, c = 6. The discriminant is (−5)² − 4·1·6 = 25 − 24 = 1, so the roots are (5 ± 1)/2 = 3 and 2. The equation factors as (x − 3)(x − 2), which is a good hand-check: multiplying it back gives exactly the original trinomial.
Example 2 — A thrown ball. A ball launched upward from ground level follows height h = −16t² + 48t, where h is in feet and t in seconds. Set h = 0 to find when it lands: −16t² + 48t + 0 = 0, so a = −16, b = 48, c = 0. The discriminant is 48² = 2304, and the roots are t = 0 (launch) and t = 3 seconds (landing). Because a is negative, the parabola opens downward and its vertex at t = 1.5 s is the maximum height, about 36 feet.
Example 3 — A profit model. Suppose a small café's monthly profit is modeled by P = −2x² + 100x − 200 dollars, where x is the price increase in dollars. To find which price increases give exactly zero profit (break-even), solve −2x² + 100x − 200 = 0. The discriminant is 10,000 − 1,600 = 8,400, giving break-even points near x = $2.08 and $47.92. Between those two price increases the café makes money; beyond either end it loses — the parabola opens downward and peaks near a $25 price increase, at roughly $1,050 profit.
Example 4 — Complex roots. Solve x² + 2x + 5 = 0: a = 1, b = 2, c = 5. The discriminant is 4 − 20 = −16, which is negative, so there are no real x-intercepts. The calculator returns the complex roots −1 + 2i and −1 − 2i. Engineering students meet these constantly: the characteristic equations of damped springs, RLC circuits, and feedback loops all live in complex plane territory.
Example 5 — A large-scale check. Solve x² + 1,000,000x = 0: a = 1, b = 1,000,000, c = 0. The roots are trivially 0 and −1,000,000, but the discriminant is 10¹². The engine computes this magnitude without rounding errors, which matters when you work with coefficients in scientific notation — a 5% error on a million-sized coefficient would move your answer by tens of thousands.
FAQ
What is the quadratic formula?
The quadratic formula solves any equation of the form ax² + bx + c = 0 in one line: x = (−b ± √(b² − 4ac)) / (2a). The ± sign gives the two roots: one using the plus sign and one using the minus sign. This calculator applies that formula directly, using complex-number arithmetic when the discriminant is negative.
What does the discriminant tell me?
The discriminant is D = b² − 4ac. If D is positive, there are two distinct real roots and the parabola crosses the x-axis twice. If D is zero, there is exactly one real root (a double root) and the parabola touches the x-axis at its vertex. If D is negative, the roots are complex conjugates and the parabola never crosses the x-axis — a quick mental filter before you even draw the graph.
Can a be negative? What happens then?
Yes. A negative a simply flips the parabola upside down: it opens downward instead of upward, so the vertex becomes a maximum rather than a minimum. The roots themselves are unchanged when you multiply the whole equation by −1; only the "opens up / opens down" flag and the meaning of the vertex flip. That is why Example 1 and the ball-throwing example work even with negative leading coefficients.
What are complex roots and why do they matter?
When the discriminant is negative, the square root of a negative number uses the imaginary unit i, where i = √(−1). The two roots come out as a pair: real part ± imaginary·i, for example −1 + 2i and −1 − 2i. They matter in physics and engineering — harmonic oscillators, AC circuits, and control systems are all analyzed in the complex plane — and in pure math they are the only "answers" that exist for such equations, even though no real x crosses the axis.
How do I find the vertex of a parabola?
The vertex x-coordinate is −b/(2a), the exact midpoint between the two roots. Plugging that x back into ax² + bx + c gives the vertex y-value, c − b²/(4a). This calculator outputs both automatically. In word problems, the vertex is usually what the question is really asking for: the maximum height, the minimum cost, or the optimal price.
What if a is zero?
Then you do not have a quadratic equation at all — you have a linear equation bx + c = 0, with the single solution x = −c/b. This calculator rejects a = 0 with an error message, because the quadratic formula divides by 2a and dividing by zero would produce nonsense. If your real problem is linear, just solve bx + c = 0 directly.
Disclaimer
This quadratic calculator is provided for informational purposes only and does not constitute mathematical, engineering, or academic advice. Roots, discriminants, and vertex values are computed from the coefficients you enter using the standard quadratic formula; while the engine is tested against reference implementations, you should verify critical results — especially complex roots and very large coefficients — with your own work or a trusted tool before relying on them for exams, engineering design, or financial decisions.