Quadratic Formula Calculator

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

By Konstantin Iakovlev · Updated September 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

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Disclaimer: This calculator is for informational purposes only. Results are estimates based on the information you provide and the assumptions described on this page.

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 use them to model parabolic paths such as a thrown object's height over time, while economists use the same parabolas to find break-even points and the output level that maximizes profit.

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: Break-Even Points of a Profit Curve

  1. 1 A small business models monthly profit, in thousands of dollars, as P(x) = -0.5x² + 10x - 20, where x is thousands of units sold. Setting P(x) = 0 finds the break-even points, so a = -0.5, b = 10 and c = -20.
  2. 2 Discriminant: Δ = (10)² - 4(-0.5)(-20) = 100 - 40 = 60. Roots: x = [-10 ± √60] / [2(-0.5)] = [-10 ± 7.746] / -1.
  3. 3 x₁ = (-10 + 7.746) / -1 = 2.254 and x₂ = (-10 - 7.746) / -1 = 17.746; the calculator shows 2.2540 and 17.7460. The vertex is at x = -10 / [2(-0.5)] = 10, where P(10) = -0.5(100) + 100 - 20 = 30, shown as (10.0000, 30.0000); the parabola opens downward.
  4. 4 The business breaks even at about 2,254 and 17,746 units a month, makes money between those two levels, and earns the most, $30,000 a month, at 10,000 units.

Source: Khan Academy · Last updated: September 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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