The quadratic equation ` x^(2) - (m -3)x + m =0` has

Options

A. Real distinct roots if and only if m ` in ( -oo, 1) cup ( 9,oo)`
B. Both positive roots if and if and only if ` m in (9,oo)`
C. Both negative roots if and only if ` m in ( 0,1)`
D. No roots

Answer

To solve the quadratic equation \( x^2 - (m - 3)x + m = 0 \) and determine the conditions for its roots, we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \).
Here,
- \( a = 1 \)
- \( b = -(m - 3) = 3 - m \)
- \( c = m \)

### Step 2: Calculate the discriminant
The discriminant \( D \) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (3 - m)^2 - 4(1)(m) = (3 - m)^2 - 4m \]

### Step 3: Expand the discriminant
Now, we will expand \( (3 - m)^2 \):
\[ D = (3 - m)(3 - m) - 4m = 9 - 6m + m^2 - 4m = m^2 - 10m + 9 \]

### Step 4: Set the discriminant greater than zero for real and distinct roots
For the quadratic equation to have real and distinct roots, the discriminant must be greater than zero:
\[ m^2 - 10m + 9 > 0 \]

### Step 5: Factor the quadratic inequality
We can factor the quadratic expression:
\[ m^2 - 10m + 9 = (m - 1)(m - 9) \]
Thus, we need to solve the inequality:
\[ (m - 1)(m - 9) > 0 \]

### Step 6: Determine the intervals
To find the intervals where the product is positive, we can analyze the critical points \( m = 1 \) and \( m = 9 \). The sign of the product will change at these points. Testing intervals:
1. For \( m < 1 \): both factors are negative, so the product is positive.
2. For \( 1 < m < 9 \): one factor is positive and the other is negative, so the product is negative.
3. For \( m > 9 \): both factors are positive, so the product is positive.

Thus, the solution to the inequality is:
\[ m < 1 \quad \text{or} \quad m > 9 \]

### Step 7: Check conditions for positive roots
For both roots to be positive, we need:
1. The sum of the roots \( \alpha + \beta = m - 3 > 0 \) implies \( m > 3 \).
2. The product of the roots \( \alpha \beta = m > 0 \) implies \( m > 0 \).

Combining these conditions with the previous intervals, we find:
\[ m > 9 \]

### Step 8: Check conditions for negative roots
For both roots to be negative:
1. The sum of the roots \( \alpha + \beta = m - 3 < 0 \) implies \( m < 3 \).
2. The product of the roots \( \alpha \beta = m > 0 \) implies \( m > 0 \).

Thus, the condition for both roots to be negative is:
\[ 0 < m < 3 \]

### Conclusion
From the analysis:
- For real and distinct roots: \( m < 1 \) or \( m > 9 \)
- For both roots positive: \( m > 9 \)
- For both roots negative: \( 0 < m < 3 \)