Question

If the roots of the equation x² – 2ax + a² + a – 3 = 0 are real and less than 3, then which one of the following is correct ?

(a) a < 2 

(b) 2 < a < 3 

(c) 3 < a < 4 

(d) a > 4

Answer 1

(a)  a < 2.

If the roots of the equation x² – 2ax + a² – a – 3 = 0 are real and less than 3, then D ≥ 0 and f (3) > 0. 

⇒ 4a² – 4(a² + a – 3) ≥ 0 and (3)² – 2a (3) + a² + a – 3 > 0 

⇒ a² – a² – a + 3 ≥ 0 and 9 – 6a + a² + a – 3 > 0 

⇒ –a + 3 ≥ 0 and a² – 5a + 6 > 0 ⇒ a – 3 ≤ 0 and (a – 2) (a – 3) > 0 

⇒ a ≤ 3 and a < 2 or a > 3 ⇒ a < 2.