Quadratic Formula Calculator

Solve quadratic equations ax²+bx+c=0. Find roots, discriminant, and vertex.

By Konstantin Iakovlev · Updated April 2026 · Source: Khan Academy

Root 1 (x₁)

3.0000

Root 2 (x₂)

2.0000

Details

Discriminant (b²−4ac)1.0000
Root TypeTwo real roots
Vertex(2.5000, -0.2500)
ParabolaOpens upward

Use the Quadratic Formula Calculator above to calculate your results. Enter your values and see instant results — all calculations run in your browser.

Disclaimer: This calculator is for informational purposes only and does not constitute tax, financial, or legal advice. Results are estimates based on the information you provide and current rates. Always consult a qualified tax professional or financial advisor for advice specific to your situation.

How It Works

Equations shaped like ax²+bx+c=0 turn up everywhere, and solving them yields the roots (x-intercepts), the discriminant, and the vertex in one pass. Engineers lean on that precision when optimizing parabolic flight paths, including the drone deliveries projected to reach 1.3 billion packages globally in 2026, while economists use the same parabolas to locate market equilibrium points and chart growth trends.

Roots come from the quadratic formula, x = [-b ± sqrt(b² - 4ac)] / 2a. The discriminant, Δ = b² - 4ac, tells you what kind of roots to expect: a positive value gives two real roots, zero gives a single real root, and a negative value gives two complex conjugate roots. The vertex, which marks the parabola's high or low point, sits at x = -b/2a, with its y-coordinate found by evaluating f(-b/2a).

One input rule governs the rest: 'a' cannot be zero, or the equation collapses into a linear one and stops being quadratic. Sign slips are the usual trap, particularly when 'b' is negative, as is dismissing complex roots as 'no solution.' Those complex roots are genuine answers and carry real weight in electrical engineering, where they describe oscillating circuits.

Example: Optimizing Drone Delivery Trajectory for a $1,500 Package

  1. 1 A drone's altitude (y) over time (x) is modeled by the equation -0.5x² + 10x - 20 = 0, where 'a' = -0.5, 'b' = 10, 'c' = -20.
  2. 2 Discriminant: Δ = (10)² - 4(-0.5)(-20) = 100 - 40 = 60. Roots: x = [-10 ± sqrt(60)] / [2(-0.5)] = [-10 ± 7.746] / -1.
  3. 3 The two real roots are x₁ = (-10 + 7.746) / -1 = 2.254 and x₂ = (-10 - 7.746) / -1 = 17.746. The vertex x-coordinate is -10 / [2(-0.5)] = 10.
  4. 4 The drone reaches its maximum altitude at 10 seconds, and touches the ground (y=0) at approximately 2.25 seconds and 17.75 seconds. This information is critical for ensuring a safe delivery window for a $1,500 package, avoiding obstacles, and optimizing fuel consumption.

Source: Khan Academy · Last updated: April 2026

Frequently Asked Questions

What is the quadratic formula?
The quadratic formula is x = (-b ± sqrt(b²-4ac)) / 2a, which solves any equation in the form ax²+bx+c = 0. The ± means there are usually two solutions (roots).
What does the discriminant tell you?
The discriminant (b²-4ac) determines the number of real solutions. If positive, there are two real roots. If zero, there is one repeated root. If negative, there are two complex (imaginary) roots.
How do you find the vertex of a parabola?
The vertex x-coordinate is -b/(2a). Plug this back into the equation to find the y-coordinate. The vertex is the maximum point if a is negative or the minimum point if a is positive.

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