Quadratic Equation with The Given Roots Calculator

A quadratic equation is a second-degree polynomial equation in a single variable. When you know the roots of a quadratic equation, you can find the equation itself using the given roots formula. This calculator helps you determine the quadratic equation when you have its two roots.

Introduction

Quadratic equations are fundamental in algebra and appear in many real-world problems. The general form of a quadratic equation is:

ax² + bx + c = 0

If you know the roots (solutions) of the equation, you can find the coefficients a, b, and c. This is useful when you have the solutions to a problem but need the original equation.

Formula

Given two roots, r₁ and r₂, of a quadratic equation, the standard form of the quadratic equation can be written as:

(x - r₁)(x - r₂) = 0

Expanding this expression gives the standard quadratic equation:

x² - (r₁ + r₂)x + (r₁ × r₂) = 0

This formula allows you to construct the quadratic equation from its roots.

How to Use the Calculator

Enter the first root (r₁) in the first input field.
Enter the second root (r₂) in the second input field.
Click the "Calculate" button to generate the quadratic equation.
The result will display the quadratic equation in standard form.

Note: The calculator assumes the leading coefficient (a) is 1. If you need a different leading coefficient, you can multiply the entire equation by that coefficient.

Example Calculation

Let's find the quadratic equation with roots 3 and -2.

Enter 3 as the first root (r₁).
Enter -2 as the second root (r₂).
Click "Calculate".

The calculator will produce the equation:

x² + (3 - 2)x + (3 × -2) = 0
x² + x - 6 = 0

This is the quadratic equation with roots 3 and -2.

FAQ

What is a quadratic equation?: A quadratic equation is a second-degree polynomial equation in a single variable, typically written in the form ax² + bx + c = 0.
How do I find the roots of a quadratic equation?: You can find the roots using the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a).
Can the roots be complex numbers?: Yes, quadratic equations can have complex roots when the discriminant (b² - 4ac) is negative.
What if the roots are equal?: If the roots are equal (a repeated root), the quadratic equation can be written as (x - r)² = 0, which expands to x² - 2rx + r² = 0.
How do I verify the result?: You can verify the result by plugging the roots back into the equation to ensure they satisfy it.