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Free online quadratic formula calculator with step-by-step solutions, interactive graphs, and detailed explanations. Solve any quadratic equation instantly!
Cannot be zero
A quadratic equation is a polynomial equation of degree 2 in the form ax² + bx + c = 0. It forms a parabola when graphed and has at most two real solutions (roots or zeros).
The vertex is the highest or lowest point on the parabola. It represents the maximum or minimum value of the quadratic function and is located at x = -b/(2a).
Quadratics model projectile motion, profit optimization, area problems, and many physics and engineering applications like bridge design and satellite dishes.
Most common form. Easy to identify coefficients for the quadratic formula.
Shows vertex (h, k) directly. Useful for graphing and optimization.
Shows roots r₁ and r₂ directly. Easiest for finding x-intercepts.
Projectile motion, free fall, and trajectory calculations
Profit optimization, cost analysis, and revenue modeling
Bridge design, arch structures, and satellite dishes
Computer graphics, animation curves, and game physics
Write your equation in standard form ax² + bx + c = 0 and identify the values of a, b, and c.
Compute Δ = b² - 4ac to determine the nature of the roots (real or complex).
Substitute a, b, and c into x = (-b ± √(b²-4ac)) / (2a) and calculate both roots.
Calculate vertex using h = -b/(2a), find axis of symmetry, and determine other key properties.
Check your solutions by substituting back into the original equation and graph the parabola.