If both roots of the equation `x^(2)-(m-3)x+m=0(m epsilonR)` are positive, then

Options

A. `m epsilon (3,oo)`
B. `m epsilon(-oo,1]`
C. `m epsilon[9,oo)`
D. `m epsilon(1,3)`

Answer

To solve the problem, we need to analyze the quadratic equation given by:

\[ x^2 - (m - 3)x + m = 0 \]

We want to find the conditions under which both roots of this equation are positive.

### Step 1: Determine the Discriminant
For the roots to be real, the discriminant (D) must be non-negative. The discriminant for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

Here, \( a = 1 \), \( b = -(m - 3) \), and \( c = m \). Thus, we have:

\[ D = (m - 3)^2 - 4(1)(m) \]

### Step 2: Set Up the Inequality
We need to ensure that the discriminant is non-negative:

\[ (m - 3)^2 - 4m \geq 0 \]

Expanding this gives:

\[ m^2 - 6m + 9 - 4m \geq 0 \]

This simplifies to:

\[ m^2 - 10m + 9 \geq 0 \]

### Step 3: Factor the Quadratic
Next, we factor the quadratic:

\[ m^2 - 10m + 9 = (m - 1)(m - 9) \]

Thus, we need to solve the inequality:

\[ (m - 1)(m - 9) \geq 0 \]

### Step 4: Analyze the Inequality
To find the intervals where this inequality holds, we can use a number line. The critical points are \( m = 1 \) and \( m = 9 \).

- For \( m < 1 \): Both factors are negative, so the product is positive.
- For \( 1 < m < 9 \): One factor is positive and the other is negative, so the product is negative.
- For \( m > 9 \): Both factors are positive, so the product is positive.

Thus, the solution to the inequality is:

\[ m \in (-\infty, 1] \cup [9, \infty) \]

### Step 5: Ensure Both Roots are Positive
Next, we need to ensure that both roots are positive. The sum and product of the roots can help us here.

1. **Sum of the Roots**: The sum of the roots is given by:

\[ S = -\frac{b}{a} = m - 3 \]

For both roots to be positive, we need:

\[ m - 3 > 0 \Rightarrow m > 3 \]

2. **Product of the Roots**: The product of the roots is given by:

\[ P = \frac{c}{a} = m \]

For both roots to be positive, we also need:

\[ m > 0 \]

### Step 6: Combine Conditions
We now combine the conditions we found:

1. From the discriminant, we have \( m \in (-\infty, 1] \cup [9, \infty) \).
2. From the sum of roots, we have \( m > 3 \).
3. From the product of roots, we have \( m > 0 \).

The only interval that satisfies all these conditions is:

\[ m \in [9, \infty) \]

### Final Answer
Thus, the values of \( m \) for which both roots of the equation are positive is:

\[ m \in [9, \infty) \]