Q. Find the value of a for which the equation $x^4 – ax^2 + 9 = 0$ has four real and distinct roots.
I have attempted this question in the following way:
Let $x^2 = t$.
So the given equation becomes $t^2 – at + 9 = 0$ –> (Equation 1)
Since we need the roots to be real and distinct:
$D > 0$
So, $a^2 – 36 > 0$ (from Equation 1)
That implies, $a\,\epsilon\, (-\infty, -6) \cup (6,\infty)$.
But the answer given is $a\,\epsilon\, (6,\infty)$. So where am I wrong?
Also if you can help me with the intervals in which a lies for no real roots and only two real roots it would be helpful.
Best Answer
HINT: the condition $a^2-36>0$ implies that there are two real and distinct roots $t $ for equation 1, but the question talks about four real and distinct roots $x$ for the initial equation.