Quadratic Formula Calculator

The quadratic formula is one of the most important tools in algebra, providing a systematic way to solve any quadratic equation. Whether you’re a student learning algebra, an engineer solving real-world problems, or anyone needing to find the roots of a parabola, understanding the quadratic formula is essential.

What is a Quadratic Equation?

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where:

• a is the coefficient of x² (must not be zero)
• b is the coefficient of x
• c is the constant term
• x represents the unknown variable we’re solving for

The Quadratic Formula

The quadratic formula states that for any quadratic equation ax² + bx + c = 0, the solutions are:

Understanding the Discriminant

The discriminant (Δ = b² - 4ac) is the expression under the square root. It tells us crucial information about the nature of the solutions:

• Δ > 0: Two distinct real roots (parabola crosses x-axis twice)
• Δ = 0: One repeated real root (parabola touches x-axis once at vertex)
• Δ < 0: Two complex conjugate roots (parabola doesn’t cross x-axis)

Step-by-Step Solution Process

1. 1. Identify coefficients: Write your equation in standard form and identify a, b, and c
2. 2. Calculate discriminant: Compute Δ = b² - 4ac to determine root type
3. 3. Apply formula: Substitute values into x = (-b ± √Δ) / (2a)
4. 4. Find both roots: Calculate using both + and - to get x₁ and x₂
5. 5. Verify: Substitute roots back into original equation to check

Example Problem

Let’s solve x² - 5x + 6 = 0:

• Coefficients: a = 1, b = -5, c = 6
• Discriminant: Δ = (-5)² - 4(1)(6) = 25 - 24 = 1
• Since Δ > 0, we have two real roots
• x = (5 ± √1) / 2 = (5 ± 1) / 2
• x₁ = (5 + 1) / 2 = 3
• x₂ = (5 - 1) / 2 = 2
• Solution: x = 2 or x = 3

Vertex and Parabola Properties

The vertex represents the turning point of the parabola:

• Vertex x-coordinate: x = -b / (2a)
• Vertex y-coordinate: Substitute x back into equation
• Axis of symmetry: Vertical line x = -b / (2a)
• Opening direction: Upward if a > 0, downward if a < 0

Real-World Applications

• Physics: Projectile motion, calculating maximum height and range
• Business: Profit optimization, break-even analysis
• Engineering: Design of parabolic structures (bridges, satellite dishes)
• Economics: Supply and demand equilibrium
• Computer Graphics: Curve rendering and animation paths
• Biology: Population growth models

Tips for Using the Quadratic Formula

• Always write equation in standard form (ax² + bx + c = 0) first
• Be careful with signs, especially negative values of b
• Calculate discriminant first to know what type of roots to expect
• For complex roots, remember i² = -1
• Verify answers by substituting back into original equation
• Use our calculator to check your work and see step-by-step solutions