Question

For the quadratic equation $x^2$ - (t - 3) x + t = 0 (t ∈ R), the values of 't' for which both the roots are greater than 2, are

• A. [3, )
• B. (- , 3]
• C. [9, )
• D. (- , -9]

Answer 1

The correct option is C

[9, )

For both the roots to exist as real numbers, D ≥ 0

Here, a = 1, b = -(t - 3), c = t

$b^2$ - 4ac ≥ 0

${(t-3)}^2$ - 4 (t) ≥ 0

$t^2$ + 9 - 6t - 4t ≥ 0

$t^2$ - 10t + 9 ≥ 0

(t - 1) (t - 9) ≥ 0

t < 1 (or) t > 9 ⇒ t ∈ (-∞, 1] ∪ [9, ∞) ----------- (1)

Now, As a > 0, graph is upward parabola min value is at x = (-b)/2a

Here, as both roots are > 2,

From the graph, f(2) > 0 ------------ (2)

and 2 < (-b)/2a ----------- (3)

(2) ⇒ 4 - (t - 3) 2 + t > 0

⇒ t < 10 ----------- (4)

(3) ⇒ 2 < (t-3)/2

⇒ t > 7 ------------- (5)

From (1), (4) & (5)

t $\in$ [9, $\infty$ )