Vieta\u2019s Formulas Calculator

This Vieta\u2019s formulas calculator gives the sum and product of the roots of a quadratic directly from its coefficients, without ever solving the equation. Enter a, b and c for a quadratic ax squared plus bx plus c, and the calculator returns the sum of the roots, the product of the roots, and the discriminant. Vieta\u2019s formulas connect the coefficients of a polynomial to symmetric functions of its roots: for a quadratic the roots add to minus b over a and multiply to c over a. They are handy for checking solutions and for building equations with chosen roots. The formulas, a worked example and the assumptions are below.

Product of roots

Discriminant

The formula

For ax^2 + bx + c = 0 with roots r and s, Vieta\u2019s formulas give r + s = minus b/a and r times s = c/a. The discriminant b^2 minus 4ac tells you whether the roots are real and distinct, repeated, or complex.

Worked example

For x^2 minus 5x + 6 (roots 2 and 3): the sum is 5, equal to minus (minus 5)/1, and the product is 6, equal to 6/1. The discriminant is 25 minus 24 = 1. Enter 1, -5, 6 to confirm.

Frequently asked questions

What are Vieta\u2019s formulas?

Relations between a polynomial\u2019s coefficients and its roots. For a quadratic, the roots sum to minus b/a and multiply to c/a.

Do I need to solve the quadratic first?

No. That is the point: Vieta gives the sum and product straight from the coefficients.

What does the discriminant tell me?

b squared minus 4ac: positive means two real roots, zero a repeated root, negative a complex conjugate pair.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

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