Fundamentals of Business Mathematics and Statistics · Quadratic Equations

Sum and Product of Roots of a Quadratic Equation

Updated 10 October 2026 · Fact-checked

For ax² + bx + c = 0 with roots α and β, the sum of roots is α + β = −b/a and the product is αβ = c/a. You use these to find expressions like α² + β² without solving, and to build an equation: x² − (sum)x + product = 0.

Understand Sum and Product of Roots

A quadratic equation ax² + bx + c = 0 (a ≠ 0) has two roots, called α and β. Usually you find them by factorising or using the formula. But many exam questions never ask for the roots themselves. They ask for something built from them, such as α² + β².

The roots are linked to the coefficients. If α and β are the roots, the equation can be written as a(x − α)(x − β) = 0. Expand it: a[x² − (α + β)x + αβ] = 0. Compare with ax² + bx + c = 0. Matching the x term gives α + β = −b/a. Matching the constant gives αβ = c/a.

This gives two numbers, the sum and the product, straight from the coefficients. No solving is needed. Any expression that is symmetric in α and β (it stays the same if you swap them) can be rewritten using only the sum and the product.

The same idea works in reverse. If you are given the roots, or their sum and product, you can write the equation directly as x² − (sum)x + product = 0. This is the fastest way to form an equation.

Key formulas to remember

Sum of roots: α + β = −b ÷ a; For ax² + bx + c = 0. Mind the minus sign.
Product of roots: αβ = c ÷ a; No minus sign. Always divide by a.
Form equation from roots: x² − (α + β)x + αβ = 0; Use when the equation is to be monic. Multiply by any number for other forms.
Sum of squares: α² + β² = (α + β)² − 2αβ; Most tested identity.
Difference of roots: (α − β)² = (α + β)² − 4αβ; Take the square root at the end; α − β = ±√(...).
Sum of cubes: α³ + β³ = (α + β)³ − 3αβ(α + β); Also α³ + β³ = (α + β)(α² − αβ + β²).
Sum of reciprocals: 1/α + 1/β = (α + β) ÷ αβ; Valid when αβ ≠ 0.

How to solve Sum and Product of Roots questions

Use this method for any question on expressions in roots or on forming equations.

1Write the equation in the form ax² + bx + c = 0 and note a, b and c with their signs.
2Find the sum S = −b/a and the product P = c/a.
3Rewrite the required expression using only α + β and αβ, using the identities above.
4Substitute S and P and simplify carefully with fractions and signs.
5To form an equation, find the new sum and new product of the new roots, then write x² − (new sum)x + new product = 0.
6Clear fractions by multiplying through by the common denominator if the options have integer coefficients.
7Check by comparing with the options, or test with simple numbers if time allows.

Quickest way: Sum-and-product shortcut

When to use it: Use when the question gives an equation and asks for a symmetric expression, or gives roots and asks for the equation.

Write S = −b/a and P = c/a at the top of your rough work.
For α² + β², compute S² − 2P directly.
For a new equation, compute the new sum and new product, then write x² − (sum)x + product.
Eliminate options whose x coefficient has the wrong sign or whose constant term does not match the product.
If a is not 1, remember to divide by a for both S and P.

Common mistakes in Sum and Product of Roots

Writing the sum of roots as b/a instead of −b/a.
Students remember the b/a part and drop the minus sign.
Fix: Always write S = −b/a first. For x² − 5x + 6 = 0, b = −5, so S = 5.
Forgetting to divide by a when a ≠ 1.
Students read off −b and c directly from the equation.
Fix: For 2x² − 8x + 6 = 0, S = 8/2 = 4 and P = 6/2 = 3.
Writing α² + β² = (α + β)² + 2αβ.
Confusion with the expansion of (α + β)².
Fix: (α + β)² = α² + 2αβ + β², so α² + β² = (α + β)² − 2αβ.
Writing the equation as x² + (sum)x + product = 0.
Sign of the middle term is mixed up.
Fix: The form is x² − (sum)x + product = 0. Check with roots 2 and 3: x² − 5x + 6.
Using the old sum and product for a new set of roots.
Students rush when the new roots are like 2α and 2β or α + 1 and β + 1.
Fix: Compute the sum and product of the new roots separately from S and P, then build the equation.

Worked examples

Example 1

If α and β are the roots of 2x² − 10x + 8 = 0, find the value of α² + β².

Show the solution

The equation is 2x² − 10x + 8 = 0, so a = 2, b = −10, c = 8.
Sum α + β = −b/a = 10/2 = 5.
Product αβ = c/a = 8/2 = 4.
α² + β² = (α + β)² − 2αβ = 25 − 8 = 17.

Answer: 17

Example 2

Form the quadratic equation whose roots are twice the roots of x² − 3x + 2 = 0.

Show the solution

For x² − 3x + 2 = 0, α + β = 3 and αβ = 2.
New roots are 2α and 2β.
New sum = 2(α + β) = 6.
New product = 4αβ = 8.
Equation: x² − 6x + 8 = 0.

Answer: x² − 6x + 8 = 0

Exam tips

Write S and P first. Most questions then take under a minute.
Check the sign of b and the value of a before you substitute.
For forming equations, test the options: the coefficient of x is −(sum) and the constant is the product.
Questions on roots like 2α, α + 1 or 1/α are common. Find the new sum and product, not the new roots.
There is no negative marking, so always mark an answer even if you must guess after eliminating options.

Practice questions from Quadratic Equations

Sum and Product of Roots in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CA FoundationSum and Product of Roots

Sum and Product of Roots: frequently asked questions

What is the formula for the sum and product of roots?

For ax² + bx + c = 0, the sum of roots is −b/a and the product is c/a. These hold whether the roots are real or not, provided a ≠ 0.

How do I form a quadratic equation when the roots are given?

Find the sum and the product of the given roots. Then write x² − (sum)x + product = 0. Multiply by a number if you need integer coefficients.

How do I find α² + β² without solving the equation?

Find α + β and αβ from the coefficients. Then use α² + β² = (α + β)² − 2αβ. Substitute and simplify.

Does this work if the roots are not real?

Yes. The relations come from comparing coefficients, so they hold for any roots of the equation, real or complex.