Question

Find all values of a for which the equation $x^4+(a-1)x^3+x^2+(a-1)x+1=0$ possesses at least two distinct negative roots.

• A. $a\ge \frac{3}{1}$
  
• B. $a\ge \frac{5}{4}$
• C. $a\ge \frac{4}{5}$
• D. $a\le \frac{4}{5}$

Answer 1

Answer: (A)

$a\ge \frac{3}{1}$

$x^4+(a-1)x^3+x^2+(a-1)x+1=0$

$⟹x^4+a{x}^4-x^3+x^2+ax-x+1=0$

$⟹xa(x^2+1)+x^2(x^2+1)-x(x^2+1)=-1$

$⟹(x^2+1)(ax+x^2-x)=-1$

$x^2+1=-1$

$⟹x^2=-2$

$⟹x=\pm 2i$

$ax+x^2-x=-1$

$⟹x^2+x(a-1)+1=0$

$\therefore b^2-4ac\ge 0$

$⟹(a-1{)}^2-4(1\left)\right(1)\ge 0$

$⟹a^2-2a+1-4\ge 0$

$⟹a^2-2a-3\ge 0$

$⟹a^2-3a+a-3\ge 0$

$⟹a(a-3)+1(a-3)\ge 0$

$⟹(a+1)(a-3)\ge 0$

$\therefore a\ge 3$