Finding range of an unknown in a biquadratic equation

Question

If the equation $x^4- ax^3+ 11x^2 -ax+1=0$ has four distinct positive roots then the range of $a$ is $(m, M) $. Find the value of $m+2M$.

Answer 1

Rewrite it as $x^2 +\dfrac{1}{x^2} - a\left(x+\dfrac{1}{x}\right) + 11 = 0$, and put $y = x+\dfrac{1}{x}\implies y^2- ay+9=0$, and this equation has $2$ positive distinct roots. Thus $\triangle > 0 \implies a^2-36 > 0\implies a > 6 = m$. Thus $y = \dfrac{a \pm \sqrt{a^2-36}}{2}> 2\implies a - \sqrt{a^2-36} > 4\implies (a-4)^2 > a^2-36 \implies 8a < 52 \implies a < \dfrac{13}{2} = M \implies m+2M = 6 + 2\cdot \dfrac{13}{2} = 19$.