If the roots of x2 - (a - 3)x + a = 0 are such that at least one of the root is greater than 2, then find the range of a.
[9, ∞)
x2 - (a - 3)x + a = 0
D= (a−3)2 - 4a = a2 - 10a + 9 = (a - 1)(a - 9)
Case 1: Both roots are greater than 2
(i) D≥0⇒ (a−1)(a−9)≥0 ⇒a∈(−∞, 1]∪[9, ∞)
(ii) f(2)>0⇒4−(a−3)2+a>0 ⇒ a<10
(iii) −b2a>2⇒a−32>2⇒a>7
∴ a∈ [9, 10)
Case 2 :- One root is greater than 2 and other is less than or equal to 2
(i) D≥0 ⇒(a−1)(a−9)≥0 a∈(−∞,1]∪[9, ∞)
(ii) f(2)≤0 ⇒4−(a−3)2+a≤0 →a≥10
∴ a∈[10, ∞)
So from both the cases we get, a∈[9, ∞)