If the both roots of the quadratic equation $x^2-mx+4=0$ are real and distinct and they lie in the interval $[1,5]$, then m lies in the interval:

Let the roots be a and b, with their product being ab = 4. We must evaluate the function f(x) = x^2 - mx + 4 at the interval endpoints, setting f(1) > 0 and f(5) > 0 to ensure both roots rest securely inside the interval. Solving 1 - m + 4 > 0 demands m < 5, while 25 - 5m + 4 > 0 demands m < 29/5. Since the roots are distinct, the discriminant m^2 - 16 > 0 requires m > 4 or m < -4. Intersecting these conditions yields the interval (4, 5).

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