Greetings,

$\displaystyle f(x)=x^2-(a^2+2a)x+a^3+a^2$ is a given function.

We are looking for all values of a such that at least one of the solutions of the function is within the interval [-1;1].

So, I've come up with this:

to find all relevent a values for exactly when one x is within [-1;1], we need this:

-1 <= x <= 0, which happens when f(-1)f(1) <= 0. (1)

To find all relevent values of a for when we need both solutions to be within [-1;1], we need:

D>=0 (2)

af(1)>=0 (3)

-1<= -b/2a <= 1 (4)

and we put it all from 1 to 4 in a system.

For the solution of (1), I have:

a is from (-1-sqrt5) / 2 to -1 U from (-1+sqrt5) / 2 to 1

(Sorry, I couldn't get Latex to work.. )

For (2), I believe all values of a do the job..

For (3):

a (-infinity;(-1-sqrt5)/2] U [1;+infinity]...

and finally for (4):

[-1-sqrt3;-1+sqrt3]

So, when we combine all of this, eventually, I am left with:

a [-1; (-1-sqrt5)/2]... which is not correct.

The true thing is supposed to be:

[(-1-sqrt5)/2; 1]

Thank you..

and sorry again for the lame writing...