Raman and Manoj attempted to solve a quadratic equation. Raman made a mistake…
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Raman and Manoj attempted to solve a quadratic equation. Raman made a mistake in writing down the constant term. He ended up with the roots (4, 3). Manoj made a mistake in writing down the coefficient of x. He got the roots as (3, 2). What will be the exact roots of the original quadratic equation?
- A.
(5, - 1)
- B.
(6, 1)
- C.
(6, - 1)
- D.
(1, 5)
Show answer & explanation
Correct answer: B
Concept: x2 + bx + c = 0 is the general monic quadratic. Its sum of roots equals −b and its product of roots equals c. If only ONE coefficient is copied down incorrectly, the relationship that depends on the OTHER, correctly-copied coefficient still holds true for the roots computed from the mistaken equation.
Application:
Raman's mistake was only in the constant term, so the coefficient of x he used was correct. This means the sum of his computed roots equals the true sum of roots of the original equation: 4 + 3 = 7.
Manoj's mistake was only in the coefficient of x, so the constant term he used was correct. This means the product of his computed roots equals the true product of roots of the original equation: 3 × 2 = 6.
The original equation can be reconstructed from this valid sum and valid product: x2 − 7x + 6 = 0.
Factorising: x2 − 7x + 6 = (x − 6)(x − 1) = 0, so the roots are 6 and 1.
Cross-check: 6 + 1 = 7, matching the sum obtained from Raman's roots, and 6 × 1 = 6, matching the product obtained from Manoj's roots — both invariants hold, confirming the result.
∴ The exact roots of the original quadratic equation are (6, 1).