The value of α for which the sum of the squares of the roots of the equation x 2 − ( a − 2 ) x − a − 1 = 0 assume the least value is

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​ x 2 − ( a − 2 ) x − a − 1 = 0 ⇒ α + β = a − 2 α β = − ( a + 1 ) α 2 + β 2 = ( α + β ) 2 − 2 α β = a 2 − 2 a + 6 = ( a − 1 ) 2 + 5 ⇒ a = 1. ​

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