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...

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):

`r_1,r_2=(a-2+-sqrt((a-2)^2-4(1)(-(a+1))))/2`

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

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

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

So `r_1^2+r_2^2=`

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

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

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

`=(2a^2-4a+12)/2`

`=a^2-2a+6`

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

------------------------------------------------------------------