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If α, β are the roots of ax 2 + bx + c = 0, (a ≠ 0) and a + δ, β + δ are the roots of Ax 2 + Bx + c = 0, (A ≠ 0) for some constant δ, then prove that b 2 – 4ac/a 2 = B 2 – 4AC/A 2 .

If α, β are the roots of ax2 + bx + c = 0, (a ≠ 0) and a + δ, β + δ are the roots of Ax2 + Bx + c = 0, (A ≠ 0) for some constant δ, then prove that b2 – 4ac/a2 = B2 – 4AC/A2.

Grade:upto college level

1 Answers

Navjyot Kalra
askIITians Faculty 654 Points
8 years ago
Sol. We know that,
(α – β)2 = [(α + δ) – (β + δ)]2
⇒ (α + β)2 - 4α β = (α + δ + β + δ)2 - 4(α + δ) (β + δ)
⇒ b2/a2 – 4c/a = B2/A2 – 4C/A ⇒ 4ac – b2/a2 = 4AC – B2/A2
[Hence α +β = -b/a, α β = c/a
(α + δ) (β + δ) = - B/A and (α + δ) (β + δ) = C/A]
Hence proved.

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