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A student claims to have found a quadratic polynomial with two REAL zeroes whose sum is 3 and whose product is 5. What is wrong with the claim?

Class 10Zeroes and CoefficientsAlgebra — Polynomials
  • AIt is impossible: αβ = 9 − 20 = −11, and can never be negative for real zeroes
  • BNothing is wrong — the two zeroes are 1 and 2
  • CIt is impossible because the product of two zeroes can never be larger than their sum
  • DIt is impossible unless the leading coefficient is taken as negative

Step-by-Step Solution

For real zeroes, must be at least 0. Here it is , which no real difference can equal — equivalently has discriminant −11. The zeroes 1 and 2 have product 2, not 5. There is no rule keeping the product below the sum: has sum 5 and product 6 and its zeroes are perfectly real. Changing the leading coefficient rescales BOTH the sum and the product, so it cannot repair the pair either.
Answer: A) It is impossible: αβ = 9 − 20 = −11, and can never be negative for real zeroes

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