Product of The Roots Calculator

What is the Product of the Roots?

A quadratic equation is any equation that can be written in the form:

Where a, b, and c are constants, and a ≠ 0. The roots (or solutions) of the quadratic equation are the values of x that satisfy the equation. The product of the roots refers to the multiplication of these two root values.

For example, if the roots of a quadratic equation are 3 and 5, the product of the roots would be 3 × 5 = 15.

How to Calculate the Product of Roots

Calculating the product of the roots of a quadratic equation is straightforward once you know the coefficients a, b, and c. The product can be found using the following formula:

This formula is derived from Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots.

To use this formula:

1. Identify the coefficients a, b, and c from the quadratic equation.
2. Divide the constant term c by the coefficient of x² (a).
3. The result is the product of the roots.

The Formula

The product of the roots of a quadratic equation ax² + bx + c = 0 is given by the simple formula:

This formula works because, according to Vieta's formulas, for a quadratic equation with roots r₁ and r₂:

• Sum of roots: r₁ + r₂ = -b/a
• Product of roots: r₁ × r₂ = c/a

The formula shows that the product of the roots is equal to the constant term divided by the leading coefficient.

Worked Example

Let's calculate the product of the roots for the quadratic equation: 2x² + 5x + 3 = 0

1. Identify the coefficients: a = 2, b = 5, c = 3
2. Apply the formula: Product = c / a = 3 / 2 = 1.5

The product of the roots is 1.5. To verify this, let's find the actual roots using the quadratic formula:

Calculating the roots:

• x₁ = [-5 + √(25 - 24)] / 4 = [-5 + 1] / 4 = -1
• x₂ = [-5 - √(25 - 24)] / 4 = [-5 - 1] / 4 = -1.5

Now, multiply the roots: (-1) × (-1.5) = 1.5, which matches our earlier result.