Mr Calcu | Find real or complex roots instantly—perfect for students, tutors, and pros who need fast, accurate answers.

Effortlessly solve any quadratic equation and uncover roots with clarity. Empower your learning and feel confident in tackling math problems today.

Quadratic Equation Solver Description

What is a Quadratic Equation?

A quadratic equation is a polynomial of degree two, typically written as:

ax² + bx + c = 0

Here, a, b, and c are constants, and a ≠ 0. These equations are foundational in algebra and appear in areas such as physics, engineering, and finance.

Standard Solution Method: The Quadratic Formula

The roots of the equation are calculated using:

x = (-b ± √(b² - 4ac)) / (2a)

The value inside the square root, b² - 4ac, is called the discriminant (Δ).

• Δ > 0: Two distinct real roots
• Δ = 0: One repeated real root
• Δ < 0: Two complex conjugate roots

Step-by-Step Derivation

To derive the quadratic formula, we complete the square:

ax² + bx + c = 0
x² + (b/a)x = -c/a
x² + (b/a)x + (b²/4a²) = -c/a + (b²/4a²)
(x + b/2a)² = (b² - 4ac)/4a²
x = (-b ± √(b² - 4ac)) / (2a)

Edge Case Handling

• a = 0: Not a quadratic equation; the calculator flags this.
• Δ = 0: The root is real and repeated; this is explicitly stated.
• All coefficients zero: Leads to 0 = 0, indicating infinite solutions.
• Discriminant is a perfect square: Results are simplified to rational roots.
• Large or precise decimal coefficients: High-precision calculation is supported.

Real-World Case Studies

Case Study 1: Physics - Projectile Motion

Height of a ball thrown upward is modeled as:

h(t) = -4.9t² + 9.8t + 1.5

To find when the ball hits the ground, solve h(t) = 0. The calculator gives the times of launch and impact.

Case Study 2: Business - Profit Modeling

A business's profit model is:

P(x) = -2x² + 40x - 150

Solving P(x) = 0 provides the break-even points. The vertex indicates the number of units for maximum profit.

Why Use This Calculator?

• Instant computation of real and complex roots
• Handles decimal, fractional, and integer coefficients
• Supports symbolic discriminant interpretation
• Provides structured output and error handling
• Useful for academic, professional, and personal scenarios

Take control of your math problems—try the calculator now and get accurate results in seconds!