A quetion on quadratic Equations Please Help Me !?

Question

Find 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 assumes the least value.
Please tell me how you got the answer also !

Answer 1

Call x and the roots
x^2 +y^2 =(x+y)^2-2xy = (a-2)^2+2(a+1)=
a^2 -2a +6 = (a-1)^2 +5.
The minimum is for a =1 and it's value is 5 because for other values of a (a-1)^2 is positiv

Answer 2

If the roots are r and s, then your polynomial can be factored as (x-r)(x-s). Multiplying that out, we have
x^2-(r+s)x+rs.

In order for this to match the polynomial you gave at the beginning, we must have r+s=a-2 (so that the linear terms match) and rs=-a-1 (so that the constant terms match).

Squaring the first of these equations, we find
r^2 + 2rs + s^2 = a^2 - 4a + 4
Doubling the second, we find
2rs=-2a-2
Subracting these last two equations, we find
r^2 + s^2 = a^2 -2a +6 (now complete the square)
= a^2 -2a +1 + 5
= (a-1)^2 + 5
...which takes its minimum value of 5 when a=1.