Find the values of ‘a ‘ for which the equation …

Q: Find the values of ‘a ‘ for which the equation
(x2 + x + 2)2 – (a – 3) (x2 + x + 2) (x2 + x + 1) +( a – 4) (x2 + x +1)2 = 0 has at least one real root.

Sol: Putting x2 + x +1 = α ,

we get (α +1)2 – ( a – 3)α( α +1) + ( a – 4) α2 = 0

⇒ α (5 – a ) + 1 = 0

⇒ α =-1/(5-a)

⇒ x2 + x + 1 =1/(a-5)

$\displaystyle \frac{1}{a-5} \ge \frac{3}{4}$

⇒ a – 5 > 0 and a – 5 ≤ 4/3

⇒ 5 < a  ≤ 19/3.