Roots and Coefficients of Quadratic Equations Calculator

A quadratic equation is a second-degree polynomial equation in a single variable. The general form is ax² + bx + c = 0, where a, b, and c are coefficients. This calculator helps you find the roots (solutions) of a quadratic equation and determine the coefficients when given the roots.

Introduction

Quadratic equations are fundamental in algebra and have wide applications in physics, engineering, and economics. The roots of a quadratic equation represent the points where the graph of the equation intersects the x-axis. The coefficients determine the shape and position of the parabola.

This calculator provides two main functions:

Calculate the roots of a quadratic equation given its coefficients
Determine the coefficients of a quadratic equation given its roots

Formula

The standard form of a quadratic equation is:

ax² + bx + c = 0

The roots of the equation can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Where the discriminant (D) is calculated as:

D = b² - 4ac

The nature of the roots depends on the discriminant:

If D > 0: Two distinct real roots
If D = 0: One real root (repeated)
If D < 0: Two complex conjugate roots

When given the roots (r₁ and r₂), the coefficients can be determined using Vieta's formulas:

a = 1 (assuming the leading coefficient is 1)
b = -(r₁ + r₂)
c = r₁ × r₂

How to Use the Calculator

Select whether you want to calculate roots or coefficients
Enter the required values in the input fields
Click "Calculate" to get the results
Review the results and chart visualization
Use the "Reset" button to clear the form

Example Calculation

Let's find the roots of the quadratic equation x² - 5x + 6 = 0:

Identify the coefficients: a = 1, b = -5, c = 6
Calculate the discriminant: D = (-5)² - 4×1×6 = 25 - 24 = 1
Apply the quadratic formula:
- x₁ = [5 + √1]/2 = 3
- x₂ = [5 - √1]/2 = 2
The roots are x = 3 and x = 2

Now, let's find the coefficients for a quadratic equation with roots 3 and 2:

Use Vieta's formulas:
- a = 1
- b = -(3 + 2) = -5
- c = 3 × 2 = 6
The equation is x² - 5x + 6 = 0

Interpreting Results

The roots of a quadratic equation represent the x-intercepts of its graph. The coefficients determine the parabola's shape and position:

If a > 0, the parabola opens upwards
If a < 0, the parabola opens downwards
The vertex of the parabola is at x = -b/(2a)

Complex roots indicate that the quadratic equation has no real solutions, only complex ones.

FAQ

What is a quadratic equation?: A quadratic equation is a second-degree polynomial equation in a single variable, typically in the form ax² + bx + c = 0.
How do I find the roots of a quadratic equation?: You can use the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a).
What does the discriminant tell me about the roots?: The discriminant (b² - 4ac) indicates the nature of the roots: positive for two distinct real roots, zero for one real root, and negative for two complex roots.
Can I find coefficients from roots?: Yes, using Vieta's formulas: a = 1 (assuming the leading coefficient is 1), b = -(r₁ + r₂), and c = r₁ × r₂.
What if the discriminant is negative?: If the discriminant is negative, the quadratic equation has two complex conjugate roots, which are solutions in the complex number system.