Product and Sum of Roots Calculator

What Are the Roots of a Quadratic Equation?

The roots of a quadratic equation are the values of x that satisfy the equation ax² + bx + c = 0. These roots can be found using the quadratic formula:

For real roots to exist, the discriminant (b² - 4ac) must be non-negative. Complex roots exist when the discriminant is negative.

Sum and Product of Roots

For any quadratic equation ax² + bx + c = 0 with roots r₁ and r₂, the sum and product of the roots can be determined directly from the coefficients:

These relationships are known as Vieta's formulas and are fundamental in algebra. They allow you to find the sum and product of roots without explicitly calculating each root.

How to Use This Calculator

1. Enter the coefficients a, b, and c of your quadratic equation in the calculator.
2. Click the "Calculate" button to compute the sum and product of the roots.
3. Review the results and use them in your calculations or further analysis.
4. Use the "Reset" button to clear the inputs and start over.

The calculator will display the sum and product of the roots based on the coefficients you provide. It also shows the discriminant to indicate whether real roots exist.

The Formula Explained

The calculator uses Vieta's formulas to determine the sum and product of the roots. Here's how it works:

These formulas are derived from the factored form of the quadratic equation: a(x - r₁)(x - r₂) = ax² - a(r₁ + r₂)x + a(r₁ × r₂). Comparing coefficients gives the relationships above.

Worked Example

Let's find the sum and product of the roots for the equation 2x² - 5x + 3 = 0.

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

Therefore, the sum of the roots is 2.5 and the product is 1.5. You can verify these results by finding the actual roots using the quadratic formula.