Sum and Product of Roots Calculator
This sum and product of roots calculator gives both numbers straight from the coefficients, without solving the equation first.
Type 2x^2 - 5x + 3 = 0 and you get sum 5/2 and product 3/2 — no roots needed anywhere.
The Two Formulas
Everything below is about the roots of a quadratic: sum and product together, taken straight from the coefficients.
The roots of a quadratic, their sum and product together, come from three numbers you can already see. For ax² + bx + c = 0 with roots α and β:
α + β = −b⁄a αβ = c⁄a
That is the whole of it. The sum of roots is one division, the product of roots is another, and neither asks you to find a root first.
Notice what is missing from both: any mention of the discriminant, any square root, any decision about whether the roots are real. None of that matters here, and that is exactly why these two formulas are worth knowing.
| Equation | Sum | Product |
|---|---|---|
| x² + 5x + 6 = 0 | −5 | 6 |
| 2x² − 5x + 3 = 0 | 5⁄2 | 3⁄2 |
| x² − 2x − 8 = 0 | 2 | −8 |
| x² + 1 = 0 | 0 | 1 |
Where They Come From
They are not a rule handed down. Write a quadratic from its own roots and expand it:
a(x − α)(x − β) = ax² − a(α + β)x + aαβ
Now set that beside ax² + bx + c. Matching the x term gives b = −a(α + β), and matching the constant gives c = aαβ. Rearrange each and both formulas fall out.
Which is why they hold for every quadratic without exception — the expansion never asked what kind of numbers α and β were.
What They Are Actually For
Not for answering "find the sum and product" questions, though those exist. For checking.
Suppose you have solved 2x² − 10x + 8 = 0 and got 4 and 1. Adding gives 5; −b⁄a is 10⁄2 = 5. Multiplying gives 4; c⁄a is 8⁄2 = 4. Both agree, so the roots are right.
Two divisions against two full substitutions. It is faster, and one pass catches almost every arithmetic slip. Every student is taught to substitute their answers back; hardly any are told this is quicker.
You can hand the check to this page as well. Add , one root is 3 to an equation with no unknown in it and it will say plainly whether 3 is a root, what substituting it actually gives if it is not, and what the other root is if it is.
The Sign Everybody Loses
That minus belongs to the formula, not to b.
In x² − 2x − 8 = 0, b is negative two. So the sum is −(−2)⁄1 = +2, not −2. The two minuses cancel, and forgetting that is the single most common mistake in this topic — which is why this page flags it whenever b comes out negative.
Its partner has no minus at all. It is c⁄a and nothing else, and people misremember it precisely because there is nothing to remember.
Roots That Are Not Real Numbers
Type x² + 1 = 0 into the calculator above.
Its roots are i and −i, which are not real numbers and cannot be plotted anywhere on a number line. Yet their sum is 0 and their product is 1 — two perfectly ordinary numbers you could hand to anyone.
Textbooks state this and rarely demonstrate it. Both formulas come from expanding brackets, and expanding brackets does not care what sort of numbers are inside them. Try x² − 2 = 0 as well: the roots are ±√2, irrational, and yet the sum is 0 and the product −2. The awkward parts cancel in the sum and multiply away in the product.
Finding a Missing Coefficient
Here is the exam question these formulas exist for.
One root of x² − 5x + k = 0 is 3. Find k and the other root.
Notice where the unknown sits: in the constant, so it never appears in −b⁄a. The sum is 5 whatever k turns out to be. One root is 3, so the other is 5 − 3 = 2. Now the product: 3 × 2 = 6, and the product is c⁄a = k, so k = 6.
No formula, no discriminant, no factorising. Type x^2 - 5x + k = 0, one root is 3 above and the page walks through exactly those three lines.
The unknown does not have to be the constant. Put the k on the x² or the x and it still works — substituting the root leaves one equation in k either way.
A Worked Example, End to End
Take 3x² + 4x − 5 = 0, which does not factorise and whose roots are not pleasant.
Read off a = 3, b = 4, c = −5. Sum is −4/3. Product is −5/3. Neither took more than a moment and neither needed a root.
Now suppose you had solved it and got roots of about 0.7863 and −2.1196. Add them: −1.3333. Multiply: −1.6667. Compare with −4/3 and −5/3, and both match. Your roots are right, and you knew it in two divisions rather than two substitutions into a quadratic.
Beyond Quadratics
Nothing about the argument was specific to a squared term. For a cubic ax³ + bx² + cx + d = 0, the three roots still add to −b⁄a, and their product is −d⁄a.
One thing does change: the sign on the product flips with the degree — plus when the degree is even, minus when it is odd. Everything else is unchanged, because the argument was only ever "expand the brackets and compare". This page handles cubics and quartics for that reason.
How to Use This Sum and Product of Roots Calculator
Type the equation as written: 2x^2 - 5x + 3 = 0. The = 0 is optional, terms can be in any order, and fractional or decimal coefficients are fine.
For the missing-coefficient question, put a k where the unknown is and add , one root is 3. If instead you know both roots and want the equation, that is the opposite journey and a different page. And if you need the roots themselves rather than their sum and product, the quadratic formula calculator finds them, the discriminant calculator tells you what kind they will be before you start, and completing the square or the vertex calculator handle the turning point.
Sum and Product of Roots Calculator FAQ
What is the sum of the roots of a quadratic?
And the product?
What are Vieta's formulas?
Do I have to find the roots first?
Do they work when the roots are not real?
How do I use them to find a missing coefficient?
Does this work for cubics?
Why is my sum coming out with the wrong sign?
Both numbers come from the coefficients, so neither waits on the roots. Use them to check work you have already done — it is quicker than substituting, and it catches the mistakes substitution is meant to.
