Sum of Roots and Product of Roots Calculator

This calculator helps you find the sum and product of roots for quadratic equations. Understanding these relationships is fundamental in algebra and has practical applications in various mathematical problems.

Introduction

For a quadratic equation in the standard form:

ax² + bx + c = 0

The sum and product of the roots (α and β) can be determined using Vieta's formulas:

Sum of roots (α + β) = -b/a

Product of roots (α × β) = c/a

These formulas are derived from the factored form of the quadratic equation:

a(x - α)(x - β) = ax² - a(α + β)x + aαβ = 0

Formula

The key formulas for the sum and product of roots are:

Sum of Roots

α + β = -b/a

This shows that the sum of the roots is equal to the negative of the coefficient of x divided by the coefficient of x².

Product of Roots

α × β = c/a

The product of the roots is equal to the constant term divided by the coefficient of x².

These relationships hold true for any quadratic equation with real or complex roots.

How to Use the Calculator

Enter the coefficients a, b, and c from your quadratic equation in the form ax² + bx + c = 0.
Click the "Calculate" button to compute the sum and product of roots.
Review the results displayed in the result panel.
Use the optional chart to visualize the relationship between the coefficients and roots.
Click "Reset" to clear the form and start a new calculation.

Note: The calculator assumes a ≠ 0. If a = 0, the equation is no longer quadratic and these formulas do not apply.

Example Calculation

Let's solve the quadratic equation: 2x² - 5x + 3 = 0

Identify the coefficients: a = 2, b = -5, c = 3
Calculate the sum of roots: (-b)/a = -(-5)/2 = 5/2 = 2.5
Calculate the product of roots: c/a = 3/2 = 1.5

The sum of the roots is 2.5 and the product is 1.5. This means if the roots are α and β, then α + β = 2.5 and α × β = 1.5.

FAQ

What if the quadratic equation has complex roots?: The formulas still apply. The sum and product will be complex numbers if the discriminant (b² - 4ac) is negative.
Can these formulas be used for higher-degree polynomials?: No, Vieta's formulas specifically apply to quadratic equations. For higher-degree polynomials, different relationships exist between the roots and coefficients.
What happens if a = 0 in the quadratic equation?: The equation becomes linear (bx + c = 0), and these formulas do not apply. The sum and product of roots concepts are not defined for linear equations.
Are there any limitations to these formulas?: The formulas assume the quadratic equation is in standard form and that a ≠ 0. They do not apply to degenerate cases or equations with repeated roots.