(x^(2)+x+2)^(2)-(a-3)(x^(2)+x+1)(x^(2)+x+2)+(a-4)(x^(2)+x+1) has at least one real root then 'a' could lie anywhere in the interval(s)
Answers
Given : (x² + x + 2)² - (a - 3)(x² + x + 1)(x² + x + 2) + (a - 4)(x² + x + 1)² has atleast one real root
To Find : 'a' could lie anywhere in the interval(s)
Solution:
(x² + x + 2)² - (a - 3)(x² + x + 1)(x² + x + 2) + (a - 4)(x² + x + 1)² = 0
=> (x² + x + 2)² - (a - 4)(x² + x + 1)(x² + x + 2) - 1(x² + x + 1)(x² + x + 2) + (a - 4)(x² + x + 1)² = 0
=> (x² + x + 2)((x² + x + 2) - (a - 4)(x² + x + 1)) - (x² + x + 1)((x² + x + 2) - - (a - 4)(x² + x + 1) =0
=> (x² + x + 2) - (a - 4)(x² + x + 1)( (x² + x + 2) - (x² + x + 1)) = 0
=> (x² + x + 2) - (a - 4)(x² + x + 1) 3 = 0
=> (x²(1 - a+ 4) + x( 1 - a + 4) + 2 - a + 4 = 0
=> x²(5 - a) + x(5 - a) + 6-a = 0
will have real root if D ≥ 0 => b² - 4ac ≥ 0
=> ( 5 - a)² - 4(5-a)(6 - a) ≥ 0
=> (5 - a) (5 - a - 24 + 4a ) ≥ 0
=> (5 - a) ( 3a - 19) ≥ 0
=> (a - 5)(3a - 19) ≤ 0
=> 5 ≤ a ≤ 19/3
'a' could lie anywhere in the interval(s) 5 ≤ a ≤ 19/3
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