The sum and product of roots help you understand a quadratic equation quickly without solving it fully. Instead of using the quadratic formula every time, these shortcut formulas directly connect the roots to the coefficients of the equation. They are extremely useful in algebra, factorization, and competitive exams.
Formula for Calculating Sum and Product of Roots – Overview
| Formula | Variables | When It Is Used |
|---|---|---|
| α + β = –b / a | a = coefficient of x², b = coefficient of x | To find the sum of roots |
| αβ = c / a | c = constant term | To find the product of roots |
What is Sum and Product of Roots in Maths?
In mathematics, the sum and product of roots describe key characteristics of the solutions of a quadratic equation without finding the actual roots. Any quadratic equation of the form ax² + bx + c = 0 has two roots, usually written as α and β.
When we write the equation in factorized form:
a(x – α)(x – β) = 0,
and expand it, we get:
ax² – a(α + β)x + aαβ = 0.
By comparing this with the original equation, we obtain:
- α + β = –b/a (sum of roots)
- αβ = c/a (product of roots)
These formulas help solve many algebraic problems faster, especially when root calculation is lengthy. They are used widely in CUET, SSC, Banking, JEE, NTSE, and school-level algebra concepts.
Examples to Calculate Sum and Product of Roots
Example 1: 2x² – 5x + 3 = 0
Step 1: a = 2, b = –5, c = 3
Step 2: Sum of roots = –b/a = –(–5)/2 = 5/2
Step 3: Product of roots = c/a = 3/2
Final Answer: Sum = 5/2, Product = 3/2
Example 2: 3x² + 7x – 6 = 0
Step 1: a = 3, b = 7, c = –6
Step 2: Sum of roots = –b/a = –7/3
Step 3: Product of roots = c/a = –6/3 = –2
Final Answer: Sum = –7/3, Product = –2
FAQs About Sum and Product of Roots Formula
Q1. Why do we use sum and product of roots?
Because they help solve quadratic questions faster without calculating the roots separately.
Q2. Can we find roots using only sum and product?
Yes, by forming a new quadratic equation:
x² – (sum)x + product = 0.
Q3. Do these formulas work for complex roots?
Yes, they work even if the roots are imaginary or complex.
Q4. Where is this concept used in real life?
It is applied in physics, engineering, optimisation, and mathematical modelling.
Q5. Is this formula included in NCERT syllabus?
Yes, it is taught in Class 10 and Class 11 algebra and is important for competitive exams.