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Relationship between zeros and coefficients of a quadratic polynomial

If α and β are the zeros of the quadratic polynomial ax^2+bx+c, then α+β=-b/a and αβ=c/a, where a is not zero.

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Student-friendly explanation

This relationship is one of the most useful results in Class 10 polynomials. It helps us find the sum and product of zeros without factorising every time. It also helps in constructing polynomials and checking answers quickly. Students should remember that the coefficient signs matter carefully.

How to write this in exams

  1. 1

    Start with the exact idea

    If α and β are the zeros of the quadratic polynomial ax^2+bx+c, then α+β=-b/a and αβ=c/a, where a is not zero.

  2. 2

    Then show how to use it

    1. Compare the polynomial with ax^2+bx+c. 2. Take a, b, and c with signs carefully. 3. Use sum = -b/a and product = c/a. 4. Check by direct multiplication or factorisation if needed.

  3. 3

    Add one concrete example

    For 2x^2-7x+3, the sum of zeros is 7/2 and the product of zeros is 3/2.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to write sum as b/a. That fails because the standard formula has a negative sign in front of b/a.

Definition

If α and β are the zeros of the quadratic polynomial ax^2+bx+c, then α+β=-b/a and αβ=c/a, where a is not zero.

Example

For 2x^2-7x+3, the sum of zeros is 7/2 and the product of zeros is 3/2.

Rule to remember

For a quadratic polynomial ax^2+bx+c with zeros α and β: α+β=-b/a and αβ=c/a.

Memory hook

Sum gets the minus sign; product does not.

Examples and method

Worked example

For x^2-5x+6, the sum of zeros is 5 and the product is 6. So the zeros are 2 and 3 because 2+3=5 and 2×3=6.

Method to apply

1. Compare the polynomial with ax^2+bx+c. 2. Take a, b, and c with signs carefully. 3. Use sum = -b/a and product = c/a. 4. Check by direct multiplication or factorisation if needed.

Diagram support

A parabola with two x-intercepts can be linked to its zeros, and those zeros can then be matched with the coefficients.

How CBSE asks it

Find sum/product of zeros, verify a relation, or form a polynomial from a given pair of zeros and coefficient.

Avoid common mistakes

Common confusion

Students often forget the minus sign in the sum formula or write product as -c/a. The product is c/a, not negative c/a.

Common wrong answer

A common wrong answer is to write sum as b/a. That fails because the standard formula has a negative sign in front of b/a.

Exam tip

For ax^2+bx+c, always write sum as -b/a and product as c/a before substituting numbers.

Quick check

What are the sum and product of the zeros of ax^2+bx+c?

The sum of the zeros is -b/a and the product of the zeros is c/a, provided a is not zero.

Answer writing and exam use

1-mark answer

If α and β are the zeros of the quadratic polynomial ax^2+bx+c, then α+β=-b/a and αβ=c/a, where a is not zero.

2-mark answer

If α and β are the zeros of the quadratic polynomial ax^2+bx+c, then α+β=-b/a and αβ=c/a, where a is not zero. For a quadratic polynomial ax^2+bx+c with zeros α and β: α+β=-b/a and αβ=c/a. For 2x^2-7x+3, the sum of zeros is 7/2 and the product of zeros is 3/2.

3-mark answer

This relationship is one of the most useful results in Class 10 polynomials. It helps us find the sum and product of zeros without factorising every time. It also helps in constructing polynomials and checking answers quickly. Students should remember that the coefficient signs matter carefully. For a quadratic polynomial ax^2+bx+c with zeros α and β: α+β=-b/a and αβ=c/a. For x^2-5x+6, the sum of zeros is 5 and the product is 6. So the zeros are 2 and 3 because 2+3=5 and 2×3=6. Find sum/product of zeros, verify a relation, or form a polynomial from a given pair of zeros and coefficient. A common wrong answer is to write sum as b/a. That fails because the standard formula has a negative sign in front of b/a.
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