Sum and Product of Roots of Quadratic Equation

Sum and Product of Roots of Quadratic Equation are obtained directly from the coefficients, without solving for the roots. For ax² + bx + c = 0, the sum of roots is −b/a and their product is c/a. These relations help calculate root expressions, form equations and solve algebra problems quickly.

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What Are the Sum and Product of Roots of a Quadratic Equation?Sum and Product of Roots of Quadratic Equation Video Lessons Sum and Product of Roots of Quadratic Equation Formula & Tricks Sum and Product of Roots of Quadratic Equation Concepts Practice Sum and Product of Roots of Quadratic Equation Questions Sum and Product of Roots of Quadratic Equation Quick Quiz Sum and Product of Roots of Quadratic Equation Revision Notes Related Topic Explainers Sum and Product of Roots of Quadratic Equation FAQs

What Are the Sum and Product of Roots of a Quadratic Equation?

For a quadratic equation ax² + bx + c = 0, where a ≠ 0, the sum of its roots is −b/a and the product of its roots is c/a.

If the roots are α and β, then α + β = −b/a and αβ = c/a. These relations follow by writing the equation as a(x − α)(x − β) = 0 and comparing coefficients. For 3x² − 7x + 2 = 0, α + β = 7/3 and αβ = 2/3.

Sum and Product of Roots of Quadratic Equation Video Lessons

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Vieta’s Formulas: Sum and Product of Roots

Learn Vieta’s formulas for quadratic equations and understand how to find the sum and product of roots using the equation’s coefficients.

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Sum and Product of Roots of Quadratic Equation Formula & Tricks

Important Formulas

Sum of roots

α + β = −b/a

For ax² + bx + c = 0, divide the coefficient of x by the coefficient of x² and change its sign.

Product of roots

αβ = c/a

Divide the constant term by the coefficient of x².

Quadratic equation from roots

x² − (α + β)x + αβ = 0

If the roots are α and β, their monic quadratic equation is x² − Sx + P = 0, where S = α + β and P = αβ.

Sum of squares of roots

α² + β² = (α + β)² − 2αβ

Using S = α + β and P = αβ, this can also be written as S² − 2P.

Sum of reciprocals

1/α + 1/β = (α + β)/(αβ) = −b/c

This relation applies when αβ ≠ 0, or equivalently c ≠ 0.

Quick Tricks

Use coefficients without finding roots

For ax² + bx + c = 0, directly use −b/a for the sum and c/a for the product. Root-solving is unnecessary unless individual roots are required.

Example: For 5x² + 3x − 2 = 0, the sum is −3/5 and the product is −2/5.

Check the signs before calculating

The sum has the opposite sign of b when a is positive, while the product has the sign of c when a is positive. If the constant term is negative, the roots have opposite signs.

Example: In 2x² − 9x − 5 = 0, the sum is 9/2 and the product is −5/2, so the roots have opposite signs.

Reduce the equation only when needed

Dividing every term by a common non-zero factor does not change the roots. This can make coefficient comparison easier.

Example: For 6x² − 12x + 6 = 0, divide by 6 to get x² − 2x + 1 = 0. The sum is 2 and the product is 1.

Sum and Product of Roots of Quadratic Equation Concepts

Finding the Sum and Product from Coefficients

For ax² + bx + c = 0, the sum of roots is −b/a and the product of roots is c/a.

The coefficient a must be non-zero. The equation need not be solved to obtain these two values. In 4x² + 8x − 3 = 0, the sum is −8/4 = −2 and the product is −3/4.

Example: For 7x² − 5x + 1 = 0, α + β = 5/7 and αβ = 1/7.

Forming a Quadratic Equation from Given Roots

If the roots are α and β, the equation is x² − (α + β)x + αβ = 0.

For roots 2 and −5, the sum is −3 and the product is −10. Therefore, the equation is x² − (−3)x − 10 = 0, or x² + 3x − 10 = 0. Any non-zero multiple of this equation has the same roots.

Example: If the sum of roots is 9 and the product is 20, the equation is x² − 9x + 20 = 0.

Calculating Expressions Involving the Roots

Expressions in α and β can be rewritten using S = α + β and P = αβ.

For example, α² + β² = S² − 2P, and α³ + β³ = S³ − 3PS. If the equation is 2x² − 6x + 1 = 0, then S = 3 and P = 1/2, so α² + β² = 3² − 2(1/2) = 8.

Example: For roots of x² − 8x + 12 = 0, α² + β² = 8² − 2(12) = 40.

Relations for Transformed Roots

If the original roots are α and β, the sum and product of transformed roots can be found algebraically from S and P.

For roots α + k and β + k, the sum is S + 2k and the product is P + kS + k². For roots kα and kβ, the sum is kS and the product is k²P. For non-zero roots 1/α and 1/β, the sum is S/P and the product is 1/P.

Example: The roots of x² − 5x + 6 = 0 have S = 5 and P = 6. The roots increased by 1 have sum 7 and product 12.

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Practice Sum and Product of Roots of Quadratic Equation Questions

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1Three numbers are in the ratio 4 : 13 : 9, and their LCM is 6084. Their HCF is:→ 2A man sells a mobile phone for ₹600 and loses an amount. If he had sold it for ₹1,080, his gain would have been 4-times the former loss. The cost price (in ₹) of the mobile phone is:→ 3Find the value of r, if 25^((r - 1)) + 100 = 5^((2r - 1))→ 4The simplified value of 178 - 7 x (3 + 17) + 22 is:→ 5If the price of a product is first increased by 18% and then decreased by 22%. Find the percentage change in the price (correct to two places of decimals).→ 6In a local community vote for a new park design, Design A receives 65% of the total 2,000 votes. How many votes did Design A receive?→ 7The seven-digit number 2A3491B is divisible by 4. Find the least possible value of A+B.→ 8My Uncle buys 300 notebooks at Rs. 25 each. He sells 120 notebooks at a 20% profit, 100 notebooks at a 10% loss, and the remaining at the cost price. What is his overall profit or loss percentage on the entire purchase?→ 9A trader buys an article and sells it at a profit of 25%. The buyer again sells it at a loss of 10%. If the final selling price is ₹9,900, then what was the cost price (in ₹) of the article for the first trader?→ 10The sum of the first pair of two-digit twin prime numbers is:→

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Quick Revision Notes

Sum and Product of Roots: Revision Points

Use these coefficient relations and transformation rules for quick calculations.

For ax² + bx + c = 0, α + β = −b/a.
For ax² + bx + c = 0, αβ = c/a.
The equation with roots α and β is x² − (α + β)x + αβ = 0.
α² + β² = (α + β)² − 2αβ.
α³ + β³ = (α + β)³ − 3αβ(α + β).
For roots α + k and β + k, the sum is S + 2k and the product is P + kS + k².
If c = 0, one root is zero, so reciprocal-root expressions are undefined.

Sum and Product of Roots of Quadratic Equation FAQs

What is the sum of roots of ax² + bx + c = 0?

The sum of roots is −b/a.

What is the product of roots of ax² + bx + c = 0?

The product of roots is c/a.

What are the sum and product of roots of 3x² − 10x + 4 = 0?

The sum is 10/3 and the product is 4/3.

How is a quadratic equation formed when the roots are 4 and 7?

The sum is 11 and the product is 28. Hence, the equation is x² − 11x + 28 = 0.

If the sum and product of roots are 6 and 5, what is the equation?

Using x² − Sx + P = 0, the equation is x² − 6x + 5 = 0.

How can α² + β² be found from the sum and product of roots?

Use α² + β² = (α + β)² − 2αβ. If the sum is 7 and the product is 10, the value is 49 − 20 = 29.

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