Q. The value of `a` 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:- A)2 B)3 C)0 D)1

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We are given the equation `x^2-(a-2)x-a-1=0` and we are asked to find the value of a that minimizes the sum of the square of the roots:

We use the quadratic formula to find the roots with a=1,b=-(a-2), and c=-(a+1):




`=(a-2)/2 +- sqrt(a^2+8)/2`

So `r_1^2+r_2^2=`


`=((a-2)^2)/4+2(a-2)/2 sqrt(a^2+8)/2+(a^2+8)/4`

`+(a-2)^2/4-2(a-2)/2 sqrt(a^2+8)/2+(a^2+8)/4`





This is minimized at a=1 (The graph of `y=a^2-2a+6` is a parabola opening up; the minimum occurs at the vertex.)


The minimum of the sum of the squares of the roots occurs when a=1