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Question
If the difference of the roots of the equation 2x2 − (a + 1)x + a − 1 = 0 is equal to their product, then prove that a = 2
Solution
2x2 − (a + 1)x + (a − 1) = 0
Let α + β be the roots
⇒ α + β = `(α + 1)/2` and αβ = `(α - 1)/2`
⇒ (α + β)2 = (αβ)2
(i.e.,) (α + β)2 – 4αβ = (αβ)2
⇒ `((α + 1)/2)^2 4((α - 1)/2) = ((α - 1)/2)^2`
`(α^2 + 2α + 1)/4 - 2(α - 1) = (α^2 - 2α+ 1)/4`
α2 + 2α + 1 − 8(α − 1) = α2 − 2α + 1
α2 + 2α + 1 − 8α + 8 − α2 + 2α − 1 = 0
− 4a + 8 = 0 ⇒ 4a = 8
α = `8/4` = 2
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