Quadratic Formula Calculator

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

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

Our Quadratic Formula Calculator precisely solves equations of the form ax²+bx+c=0, revealing the roots (x-intercepts), discriminant, and vertex. This is crucial for fields like engineering, where optimizing parabolic trajectories for drone deliveries in 2026, projected to reach 1.3 billion packages globally, requires pinpoint accuracy. Understanding these parameters also aids economists in modeling market equilibrium points and predicting growth trends.

The calculator utilizes the quadratic formula, x = [-b ± sqrt(b² - 4ac)] / 2a, to determine the roots. The discriminant, Δ = b² - 4ac, indicates the nature of these roots: positive for two real roots, zero for one real root, and negative for two complex conjugate roots. The vertex, representing the parabola's maximum or minimum, is found at x = -b/2a and y = f(-b/2a).

When inputting coefficients, ensure 'a' is not zero, as this would result in a linear equation, not a quadratic. Common mistakes include sign errors within the formula, especially with negative 'b' values, or misinterpreting complex roots as non-solutions. Remember that complex roots are valid solutions, often significant in electrical engineering applications involving 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.

You might also need

Percentage Calculator

Calculate percentages three ways: what is X% of Y, X is what % of Y, and percentage change from X to Y.

Fraction Calculator

Add, subtract, multiply, and divide fractions. Results shown as fraction, mixed number, and decimal.

Standard Deviation Calculator

Calculate mean, variance, and standard deviation (population and sample) from a data set.