Quadratic Calculator - Solve Quadratic Equations Online

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2, typically written in the standard form:

ax² + bx + c = 0

Where a, b, and c are coefficients, and a ≠ 0. The term "quadratic" comes from the Latin word "quadratus," meaning "square," because the highest power of the variable is 2 (squared).

Key Components of Quadratic Equations

Coefficient a: The coefficient of x² (must not be zero)
Coefficient b: The coefficient of x
Coefficient c: The constant term
Variable x: The unknown value we're solving for

The Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation. It provides the solutions (roots) of the equation directly:

x = (-b ± √(b² - 4ac)) / 2a

Understanding the Discriminant

The expression under the square root, b² - 4ac, is called the discriminant (Δ). It determines the nature of the roots:

Δ > 0

Two real and distinct roots

Δ = 0

One real root (repeated)

Δ < 0

Two complex conjugate roots

Step-by-Step Solution Process

Identify coefficients: Extract a, b, and c from the equation
Calculate discriminant: Compute Δ = b² - 4ac
Determine nature of roots: Based on the discriminant value
Apply quadratic formula: Substitute values into the formula
Simplify results: Calculate the final root values

Common Examples

Example 1: x² - 5x + 6 = 0

a = 1, b = -5, c = 6

Discriminant: 25 - 24 = 1 > 0 (Two real roots)

Roots: x = 2 and x = 3

Example 2: x² - 4x + 4 = 0

a = 1, b = -4, c = 4

Discriminant: 16 - 16 = 0 (One repeated root)

Root: x = 2 (repeated)

Example 3: x² + 2x + 5 = 0

a = 1, b = 2, c = 5

Discriminant: 4 - 20 = -16 < 0 (Complex roots)

Roots: x = -1 ± 2i

Applications of Quadratic Equations

Quadratic equations have numerous real-world applications across various fields:

Physics: Projectile motion, parabolic trajectories
Engineering: Structural analysis, optimization problems
Economics: Profit maximization, cost analysis
Geometry: Area and perimeter calculations
Computer Science: Algorithm complexity analysis
Biology: Population growth models

Alternative Solution Methods

1. Factoring

When possible, factor the quadratic expression into two binomials. This method works well when the equation has integer roots.

2. Completing the Square

Transform the equation into the form (x + h)² = k, then solve for x. This method is particularly useful for deriving the quadratic formula.

3. Graphing

Plot the quadratic function y = ax² + bx + c and find where it intersects the x-axis. The x-intercepts are the roots of the equation.

Tips for Using the Calculator

Enter coefficients as decimal numbers (e.g., 2.5, -3.14)
Use negative signs for negative coefficients
The coefficient 'a' must not be zero
Results are displayed with 6 decimal places for precision
Complex roots are shown in the form a + bi
Use the step-by-step solution to understand the process

Frequently Asked Questions

What happens if the coefficient 'a' is zero?

If 'a' is zero, the equation becomes linear (bx + c = 0), not quadratic. Our calculator will show an error message because the quadratic formula requires division by 2a, which would be undefined if a = 0.

How do I interpret complex roots?

Complex roots occur when the discriminant is negative. They appear in the form a + bi and a - bi, where 'a' is the real part and 'b' is the imaginary part. These roots are conjugates of each other and are equally valid solutions.

Can I solve quadratic equations with fractions?

Yes, you can enter fractional coefficients as decimals. For example, if you have 1/2x² + 3/4x - 1/8 = 0, enter a = 0.5, b = 0.75, and c = -0.125.

What's the difference between real and complex roots?

Real roots are actual numbers that can be plotted on a number line. Complex roots involve imaginary numbers and cannot be represented on a traditional number line. Real roots correspond to x-intercepts of the parabola, while complex roots mean the parabola doesn't cross the x-axis.

How accurate are the calculator results?

The calculator provides results accurate to 6 decimal places. For most practical purposes, this level of precision is sufficient. If you need higher precision, you may need specialized mathematical software.

Can I use this calculator for quadratic inequalities?

This calculator solves quadratic equations (equality), not inequalities. However, knowing the roots helps identify key points for solving inequalities. The roots divide the number line into intervals that can be tested for inequality solutions.