Vieta’s Formulas Calculator

This Vieta’s formulas calculator finds the sum and product of the roots of any quadratic straight from its coefficients, without you ever having to solve the equation. Vieta’s formulas connect the coefficients of a polynomial to symmetric functions of its roots: for a quadratic written as ax squared plus bx plus c, the two roots always add to minus b over a, and always multiply to c over a. You enter a, b and c into the three fields, and the calculator instantly returns the sum of the roots, the product of the roots, and the discriminant, b squared minus 4ac. The discriminant is worth watching, because its sign shows what kind of roots you have: positive means two distinct real roots, zero means a single repeated root, and negative means a pair of complex conjugate roots. This makes the tool useful well beyond one calculation. Algebra students can use it to check answers after factorising or applying the quadratic formula by hand, teachers can use it to build examples quickly, and anyone constructing a quadratic with chosen roots can work backwards from a target sum and product to find the coefficients needed. Because Vieta’s whole point is skipping the solving step, this tool works best when you already know a, b and c and just want the relationships confirmed. A worked example and formula are set out below.

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Sum of roots = 5

Product of roots6

Discriminant1

The formula

For ax^2 + bx + c = 0 with roots r and s, Vieta’s 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’s formulas?

Relations between a polynomial’s 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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