Free Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 instantly. Enter the three coefficients to get real or complex roots, the discriminant, vertex coordinates, and axis of symmetry. Free, private — all calculations run in your browser.
Quadratic Formula Calculator — Solve ax²+bx+c=0
Solve any quadratic equation of the form ax²+bx+c = 0 instantly. Enter coefficients a, b and c to find real or complex roots, the discriminant, vertex, and axis of symmetry. Ideal for algebra students, engineers and physicists solving polynomial equations.
How to Use This Calculator
- 1
Enter the coefficient "a" (the number multiplying x²). Must not be zero.
- 2
Enter coefficient "b" (the number multiplying x). Can be negative.
- 3
Enter coefficient "c" (the constant term). Can be zero.
- 4
Roots, discriminant, vertex and axis of symmetry appear instantly.
- 5
Export a PDF report with your full solution.
📐 How This Is Calculated
x = (−b ± √(b² − 4ac)) / 2a
a—Coefficient of x² — determines parabola direction (a>0 opens up, a<0 opens down)b—Coefficient of xc—Constant termΔ = b²−4ac—Discriminant: Δ>0 → 2 real roots; Δ=0 → 1 repeated root; Δ<0 → complex rootsReference: Elementary algebra — Al-Khwarizmi (9th century); modern form from Vieta's Formulas
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Results are provided for informational and educational purposes only. They should not be used as a substitute for professional financial, engineering, medical, or legal advice. Always verify outputs with a qualified professional before making important decisions. Roughtools makes no warranties regarding accuracy or completeness for your specific situation.
About This Quadratic Formula Calculator
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. Quadratic equations are among the most important in mathematics — they model projectile trajectories, areas, profit maximisation, structural loads, electrical circuits, and countless other real-world phenomena. This calculator solves any quadratic equation instantly using the quadratic formula and provides all related values: discriminant, both roots (real or complex), vertex, and axis of symmetry.
The Quadratic Formula
For the equation ax² + bx + c = 0, the two solutions (roots) are:
Discriminant D = b² − 4ac
The ± symbol produces two solutions: x₁ uses +√D and x₂ uses −√D. The discriminant D controls the nature of the solutions: D > 0 → two distinct real roots, D = 0 → one repeated real root (x = −b/2a), D < 0 → two complex conjugate roots.
The Discriminant Explained
The discriminant D = b² − 4ac is the expression under the square root in the quadratic formula. It "discriminates" between the three cases:
- •D > 0: two distinct real roots — the parabola crosses the x-axis at two points
- •D = 0: one repeated real root at x = −b/(2a) — the parabola is tangent to the x-axis
- •D < 0: no real roots — two complex conjugate roots — the parabola does not intersect the x-axis
Vertex Form and the Parabola
Every quadratic y = ax² + bx + c describes a parabola. The vertex is its highest or lowest point. The x-coordinate of the vertex is x = −b/(2a) — also the axis of symmetry about which the parabola is mirror-symmetric. The y-coordinate is found by substituting this x back: y = a(−b/2a)² + b(−b/2a) + c = c − b²/(4a). If a > 0, the parabola opens upward and the vertex is a minimum (optimisation: minimise cost, distance). If a < 0, the parabola opens downward and the vertex is a maximum (optimisation: maximise profit, height, area).
Complex and Imaginary Roots
When D < 0, the square root of a negative number is required. The imaginary unit i is defined as i = √(−1), so √(−D) = i√|D|. The two complex roots are: x = (−b + i√|D|) / (2a) and x = (−b − i√|D|) / (2a). These are complex conjugates — they always appear in conjugate pairs. In engineering, complex roots arise in oscillation and signal problems. In a real-world projectile or geometric problem, complex roots simply mean there is no real solution (the projectile never reaches that height, or the dimensions have no real solution).
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When to Use This Calculator
The height of a projectile over time follows a quadratic equation. Solve for when the object hits the ground (h = 0), reaches maximum height (vertex), or passes a specific height.
Revenue, profit, area, and cost functions are often quadratic. The vertex gives the maximum revenue or minimum cost. Solve ax² + bx + c = 0 to find break-even points.
When the area of a rectangle, triangle, or other shape is given as a quadratic expression in one unknown dimension, the quadratic formula gives the required dimensions directly.
Bending moment equations, cable catenary problems, and signal processing filters all produce quadratic equations. Get roots and vertex coordinates for structural analysis.
When total cost and total revenue are both expressed as functions of quantity produced, setting them equal produces a quadratic equation. Solve for the break-even quantity.
💡 Pro Tips
Always compute the discriminant D = b² − 4ac before applying the full formula. It tells you instantly how many solutions to expect: D > 0 → two real solutions, D = 0 → one solution, D < 0 → no real solutions. This check saves time and prevents confusion about why the calculator shows complex results.
If a = 0, do not use this calculator — the equation is linear, not quadratic. The solution is simply x = −c/b. Substituting a = 0 into the quadratic formula produces a division by zero. Always verify a ≠ 0 before entering coefficients.
Verify your solutions by substituting them back into the original equation. If x = r is a root of ax² + bx + c = 0, then a(r)² + b(r) + c should equal exactly 0 (or very close to 0, accounting for floating-point precision). This sanity check catches sign errors and transcription mistakes in the coefficients.
The vertex of the parabola y = ax² + bx + c is at x = −b/(2a) on the x-axis and y = c − b²/(4a) on the y-axis. The vertex is the maximum point if a < 0 (parabola opens downward) and the minimum point if a > 0 (parabola opens upward). Many optimisation problems in physics and economics reduce to finding the vertex of a quadratic.
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