# Four roots of an equation

• Oct 9th 2008, 10:26 AM
Logic
Four roots of an equation
Greetings,
This is a homework. Not quite an urgent one - I need it for monday and I have had it since monday. Unfortunately, few ideas occured to me.
Obviously, a reasonable step, perhaps, is to make a substitution, but then, I believe, we get only a link between m and y, assuming y is the substitution y = x*x -2x + 2, and no link between m and x.
The problem is taken from a university entrance test. We need this equation to have 4 different roots.
• Oct 9th 2008, 10:44 AM
icemanfan
Substitution is a good idea. You will get $y^2 - 3my + 3 = 0$, from which you can use the quadratic formula:

$y = \frac{3m \pm \sqrt{(3m)^2 - 4(1)(3)}}{2}$

$y = \frac{3m \pm \sqrt{9m^2 - 12}}{2}$

If this equation is to have any real roots at all, $9m^2 - 12 \geq 0$

$9m^2 \geq 12$

$m^2 \geq \frac{4}{3}$

Then you have $x^2 + 2x + 2 = \frac{3m \pm \sqrt{9m^2 - 12}}{2}$

$x^2 + 2x + 2 - \frac{3m \pm \sqrt{9m^2 - 12}}{2} = 0$

and you can use the quadratic formula again.