Two students A and B solve an equation of the form x2 + px + q = 0. A

1.	Two students A and B solve an equation of the form x2 + px + q = 0. A starts with a wrong value of p and obtains the roots as 2 and 6. B starts with a wrong value of q and gets the roots as 2 and –9. What are the correct roots of the equations ? (a) 3 and –4 (b) –3 and –4 (c) –3 and 4 (d) 3 and 4
Answer» (b) - 3 and - 4 Let the roots of the quadratic equation x² + px + q = 0 be α and β. According to the given condition, A starts with a wrong value of p and obtains the roots as 2 and 6. But this time, the value of q is correct. ∴ q = Product of roots = αβ = 2 × 6 = 12. According to the second condition, B starts with a wrong value of q and obtains the roots as 2 and –9. But this time, the value of p is correct. ∴ p = sum of roots = α + β = 2 + (–9) = –7 ...(i) ∴ (α - β)² = (α + β)² – 4ab = (–7)² – 4.12 = 49 – 48 = 1 ⇒ α - β = 1 ...(ii) ∴ Solving equations (i) and (ii), we get α = – 3 and β = – 4.

Two students A and B solve an equation of the form x2 + px + q = 0. A starts with a wrong value of p and obtains the roots as 2 and 6. B starts with a wrong value of q and gets the roots as 2 and –9. What are the correct roots of the equations ? (a) 3 and –4 (b) –3 and –4 (c) –3 and 4 (d) 3 and 4

Answer»

(b) - 3 and - 4

Let the roots of the quadratic equation x² + px + q = 0 be α and β. According to the given condition, A starts with a wrong value of p and obtains the roots as 2 and 6. But this time, the value of q is correct.

∴ q = Product of roots = αβ = 2 × 6 = 12.

According to the second condition, B starts with a wrong value of q and obtains the roots as 2 and –9. But this time, the value of p is correct.

∴ p = sum of roots = α + β = 2 + (–9) = –7 ...(i)

∴ (α - β)² = (α + β)² – 4ab = (–7)² – 4.12 = 49 – 48 = 1

⇒ α - β = 1 ...(ii)

∴ Solving equations (i) and (ii), we get α = – 3 and β = – 4.

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