Quadratic equation #1

[mathwannabe]

Hello everybody

I need some help.

The problem says (I hope I will translate it correctly):

1) The value of the real parameter m for which the sum of the squares of the roots of the equation x^2 - mx + m - 3 = 0 is smallest, belongs to which interval: here they give a bunch of offered answers.

OK, since I have no idea even where to begin I'm gonna ask for your help. I was trying something like: Well, if the equation has only one solution, then the sum of squares of its roots will be smallest, so discriminant should be equal to zero. The discriminant of the equation is m^2 - 4m + 12. So I set it up to be m^2 - 4m + 12 = 0 and then solve, but I get conjugate complex solution. I am completely confused. Please help...

 

[Deleted member 4993]

Hello everybody

I need some help.

The problem says (I hope I will translate it correctly):

1) The value of the real parameter m for which the sum of the squares of the roots of the equation x^2 - mx + m - 3 = 0 is smallest, belongs to which interval: here they give a bunch of offered answers.

OK, since I have no idea even where to begin I'm gonna ask for your help. I was trying something like: Well, if the equation has only one solution, then the sum of squares of its roots will be smallest, so discriminant should be equal to zero. The discriminant of the equation is m^2 - 4m + 12. So I set it up to be m^2 - 4m + 12 = 0 and then solve, but I get conjugate complex solution. I am completely confused. Please help...

This is my interpretation of the problem:

If we have a quadratic equation of the form AX² + BX + C = 0

Then sum of the squares of the roots

S = [{-B + √(B² - 4AC)}² + {-B - √(B² - 4AC)}²]/(4A²) = (B² - 2AC)/A²


Then here we have

S = m² - 2*(m-3)

This is a parabola with a vertex at (m₁, S₁). Find m₁ and S₁.

 

[pka]

1) The value of the real parameter m for which the sum of the squares of the roots of the equation x^2 - mx + m - 3 = 0 is smallest, belongs to which interval: here they give a bunch of offered answers.

This is really a calculus problem.
Suppose that \(\displaystyle a~\&~b\) the roots the you want to minimize \(\displaystyle S=a^2+b^2\).
We know that \(\displaystyle a+b=m\), sum of the roots.
Also \(\displaystyle ab=m-3\), the product of the roots.
Use those two to get \(\displaystyle S\) as a function of \(\displaystyle m\) alone.
Then minimize \(\displaystyle S\).

Here is a hint to get you started: \(\displaystyle a^2+2ab+b^2=m^2\).