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Mathematics Question on Complex Numbers and Quadratic Equations

The value of aa for which the sum of the squares of the roots of the equation x2−(a−2)x−a−1=0x^2 - (a - 2) x - a - 1 = 0 assume the least value is

A

1

B

0

C

3

D

2

Answer

1

Explanation

Solution

Let α,β\alpha, \beta be the roots of the equation ∴ α+β=a−2\therefore\, \alpha+\beta=a-2 and αβ=−(a+1)\alpha\beta=-\left(a+1\right) Now α2+β2=(α+β)2−2αβ\alpha^{2}+\beta^{2}=\left(\alpha+\beta\right)^{2}-2\alpha\beta =(a−2)2+2(a+1)=\left(a-2\right)^{2}+2\left(a+1\right) =(a−1)2+5=\left(a-1\right)^{2}+5 ∴ α2+β2\therefore\, \alpha^{2}+\beta^{2} will be minimum if (a−1)2=0\left(a-1\right)^{2}=0 , i.e., a=1a=1 .