Multiple choice

If both the roots of the equation $x^2-9x+a=0$ are positive and one is greater than $3$ and other is less than $3$, then all possible values of $a$ is

  1. $0$ $<$ $a$ $<$ $18$
  2. $-1$ $<$ $a$ $<$ $2$
  3. $-18$ $<$ $a$ $<$ $0$

  4. None of the above.
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A Correct answer
Explanation

For the roots of x^2 - 9x + a = 0 to be on opposite sides of 3, the function f(x) = x^2 - 9x + a must satisfy f(3) < 0. Calculating f(3) = 9 - 27 + a = a - 18, so a - 18 < 0 implies a < 18. Since the roots are positive, the product of roots (a) must be > 0, so 0 < a < 18.

AI explanation

Let the roots be alpha and beta, where one root is less than 3 and the other is greater than 3. This means the quadratic expression x^2 - 9x + a evaluates to a negative number at x = 3, so 3^2 - 9(3) + a < 0, which simplifies to a < 18. Since both roots are positive, their product must be positive, meaning a = (alpha)(beta) > 0. Combining these two inequalities gives 0 < a < 18.

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