The Solution
Calculation: Vieta's Formulas Calculator
Vieta's Formulas Calculator
Enter your polynomial coefficients above and click "Apply Vieta's Formulas" to see results.Graph will appear here after you solve.
Cubic Equation Solver
Vieta's Formulas solver
Find real and complex roots, discriminant, and cubic graphs in seconds with our precision solver.
The Equation
Provide coefficients to verify Vieta's relationships between the roots and the equation's coefficients.
f(x) = 0 = 0
The Treatise
What isVieta's Formulas Calculator?
- For a school student: The three answers of a cubic are secretly related. If you add them up, multiply them in pairs, and multiply all three, you get the numbers from the original equation, almost directly. Vieta's formulas are these three secret relationships, and they let you check answers without re-solving anything.
- For a college student: For ax³ + bx² + cx + d = 0 with roots r₁, r₂, r₃: r₁ + r₂ + r₃ = −b/a, r₁r₂ + r₁r₃ + r₂r₃ = c/a, r₁r₂r₃ = −d/a. They come from expanding a(x − r₁)(x − r₂)(x − r₃) and matching coefficients. The signs alternate −, +, −, which is the single most useful memory hook.
- For the advanced reader: These are the elementary symmetric polynomials e₁, e₂, e₃ in the roots, and the fundamental theorem of symmetric polynomials says every symmetric expression in the roots is a polynomial in these three. Newton's identities go further, linking power sums pₖ = r₁ᵏ + r₂ᵏ + r₃ᵏ to the coefficients. Galois theory begins here: the coefficients are exactly the symmetric functions fixed by every root permutation.
- Why it matters in practice: Vieta's formulas are the fastest sanity check in cubic work. Found candidate roots with our cubic roots calculator? Add them: the sum must equal −b/a. One addition catches most arithmetic slips. They also run in reverse: prescribe root relationships and build the equation with our factorization tool.
The Formula
Formula / Method
- The three relations: r₁ + r₂ + r₃ = −b/a, r₁r₂ + r₁r₃ + r₂r₃ = c/a, r₁r₂r₃ = −d/a.
- Derivation in one line: Expand a(x − r₁)(x − r₂)(x − r₃) = ax³ − a(Σrᵢ)x² + a(Σrᵢrⱼ)x − ar₁r₂r₃ and match against ax³ + bx² + cx + d. Each Vieta relation is one coefficient match.
- Sign mnemonic: The signs go −, +, − down the list. The odd-position relations (sum, product) carry the minus; the even one (pairwise) is positive. For monic cubics (a = 1) this is the whole story.
- A common trap: For non-monic cubics you must divide by a. The sum is −b/a, not −b. Forgetting the division is the number-one Vieta error, so always normalize first or let the tool handle it.
The Method
How To Use
- Step 1. Enter your coefficients a, b, c, d, or enter three roots to run Vieta in reverse.
- Step 2. Press "Apply Vieta's Formulas".
- Step 3. Read the three relations with their numeric values, plus the normalized monic form for clarity.
- Step 4. Use the relations to verify roots from the cubic roots calculator, or feed prescribed sums into the factorization tool to build new equations.
Key Features
- Computes all three Vieta relations instantly, with correct signs and division by a.
- Runs in reverse: enter desired root relationships and get the cubic.
- Shows the monic normalized form so the sign pattern is obvious.
- Verifies candidate roots against all three relations at once.
- Handles complex roots transparently, since the relations hold for all roots.
The Illustration
Visual Diagram
Example Concept
Take the roots 1, 2, 3. Vieta predicts the monic cubic:
Sum: 1 + 2 + 3 = 6 = −b, so b = −6.
Pairwise: (1)(2) + (1)(3) + (2)(3) = 2 + 3 + 6 = 11 = c.
Product: (1)(2)(3) = 6 = −d, so d = −6.
The equation is x³ − 6x² + 11x − 6 = 0. Expand (x − 1)(x − 2)(x − 3) to confirm it matches term by term. Reverse check with our cubic roots calculator: it returns 1, 2, 3, closing the loop.
The Deep Dive
Interactive Deep Dive
The proof is a single expansion. Write a(x − r₁)(x − r₂)(x − r₃) and multiply out: the x² coefficient collects −a(r₁ + r₂ + r₃), the x coefficient collects a(r₁r₂ + r₁r₃ + r₂r₃), and the constant term is −ar₁r₂r₃. Matching against ax³ + bx² + cx + d gives all three relations at once. This argument generalizes word for word to degree n, producing the alternating-sign pattern students meet again with quartics and beyond.
Newton's identities extend Vieta from elementary sums to power sums. With p₁ = r₁ + r₂ + r₃, p₂ = r₁² + r₂² + r₃², the identities give p₁ = −b/a and p₂ = (b/a)² − 2(c/a). For roots 1, 2, 3: p₂ = 1 + 4 + 9 = 14, and (−6)² − 2(11) = 36 − 22 = 14. These let you compute symmetric expressions like the sum of squares directly from coefficients, a staple of contest problems asking for quantities no one wants to compute root by root.
Vieta runs backward as a construction tool. Want a cubic with roots summing to 0 and product 6? The relations give b = 0, d = −6 immediately, and choosing the pairwise sum fixes c. This is how textbooks manufacture "nice" problems and how engineers impose constraints: specify the desired root relationships, read off the required coefficients. Our factorization tool then confirms the construction factors as intended.
Even the discriminant speaks Vieta's language: Δ = a⁴(r₁ − r₂)²(r₁ − r₃)²(r₂ − r₃)² is symmetric in the roots, so the fundamental theorem of symmetric polynomials guarantees it can be rewritten purely in b, c, d. That rewriting is exactly the five-term formula. Vieta's relations are thus the dictionary translating between the world of roots and the world of coefficients, with the discriminant as its most famous entry.
François Viète published these relations in 1591, before modern algebraic notation existed: he wrote them out in words. The deeper significance emerged later: the coefficients are precisely the polynomial expressions in the roots that survive every permutation of the roots. Galois theory starts from this observation, asking which non-symmetric expressions survive some permutations. Vieta's formulas are the degree-three doorway into one of mathematics' deepest theories.
The Applications
Real-World Applications
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Answer Checking
The single most common use: verify candidate roots in seconds. If your three numbers do not sum to −b/a, at least one is wrong, no substitution needed. Teachers grade with it, students self-check with it, and software uses it as a cheap assertion after the cubic roots calculator finishes.
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Contest Mathematics
Olympiad problems love asking for symmetric expressions like r₁² + r₂² + r₃² or 1/r₁ + 1/r₂ + 1/r₃ without giving the roots. Vieta plus Newton's identities answer these directly from the coefficients. Whole problem sets are built on this trick.
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Equation Design
Need a cubic with specific root properties, say roots summing to zero for a centered design? Vieta converts root specifications into coefficient requirements instantly. Control engineers use this when placing system poles to meet performance targets.
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Physics Sum Rules
In physics, sums over eigenvalues often have closed forms via Vieta-style relations on characteristic polynomials. From coupled oscillators to tight-binding models, the sum of the roots is frequently the only aggregate the theory needs, readable straight from one coefficient.
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Computer Algebra
Symbolic systems use Vieta relations to simplify expressions in roots without computing the roots themselves. Resultants, discriminants, and elimination theory all run on symmetric-polynomial machinery that starts with these three formulas.
The Cautions
Common Mistakes to Avoid
MISTAKE Nº 1
Sign errors on −b/a and −d/a: The number-one Vieta mistake. The sum is −b/a and the product is −d/a, with the minus signs. For x³ − 6x² + 11x − 6 the root sum is +6, not −6. Memorize the −, +, − pattern.
MISTAKE Nº 2
Forgetting to divide by a: The relations need the monic-normalized coefficients. For 2x³ − 12x² + …, the root sum is −(−12)/2 = 6, not 12. Either divide the equation by a first or let the tool normalize it.
MISTAKE Nº 3
Mixing up c/a: The pairwise sum r₁r₂ + r₁r₃ + r₂r₃ equals +c/a, positive. Students often attach a minus by false analogy with the other two relations. Only the odd positions (sum, product) carry the minus.
MISTAKE Nº 4
Applying to non-polynomials: Vieta's formulas are statements about polynomial coefficients. They say nothing about roots of transcendental equations like x³ = eˣ. Confirm you have a genuine polynomial first.
MISTAKE Nº 5
Assuming integer roots: The relations hold for irrational and complex roots too. A product of 6 does not force integer roots: try 1, 2, 3 versus ∛2, ∛4/… style triples. Use the formulas as constraints, not as a promise of niceness.
The Reference
Quick Reference Table
Quick Reference TableREF
Sum of Rootsr₁ + r₂ + r₃ = −b/a
Pairwise Productsr₁r₂ + r₁r₃ + r₂r₃ = c/a
Product of Rootsr₁·r₂·r₃ = −d/a
Named AfterFrançois Viète (1540–1603)
Works WithBoth real and complex roots
The Questions
Frequently Asked Questions
What are Vieta's formulas? +
Three relations linking a cubic's coefficients to its roots: the roots sum to −b/a, their pairwise products sum to c/a, and their product is −d/a.
Why do the signs alternate −, +, −? +
Because expanding a(x − r₁)(x − r₂)(x − r₃) introduces a minus with each factor. Odd-position relations inherit the minus; the even one does not.
Do they work for non-monic cubics? +
Yes, after dividing by the leading coefficient a. The relations are stated with /a precisely so they hold for every cubic.
Can I find the roots from Vieta alone? +
The three relations are three equations in three unknowns, so in principle yes, but solving that system is itself hard. In practice use them to check roots found by the cubic roots calculator or guessed via the rational root theorem.
Who was Vieta? +
François Viète (1540–1603), French mathematician who published the relations in 1591 and pioneered symbolic algebra, using letters for both unknowns and parameters.
Do similar formulas exist for higher degrees? +
Yes. For degree n there are n relations with alternating signs, each an elementary symmetric sum of the roots. The cubic case is the most used.
Do they work when roots are complex? +
Absolutely. The relations are pure algebra and hold for complex roots unchanged. Check examples with our complex roots calculator.
How do I use them to check my answers? +
Add your three candidate roots: the sum must equal −b/a. This one addition catches the majority of arithmetic slips before you ever substitute back.