Find the values of m for which the quadratic equation `x^(2)-m(2x-8)-15=0` has
(i) equal roots,
(ii) both roots positive.

Answer

To solve the quadratic equation \( x^2 - m(2x - 8) - 15 = 0 \) for the values of \( m \) under the given conditions, we will follow these steps:

### Step 1: Rewrite the Quadratic Equation
The given equation can be rewritten as:
\[ x^2 - m(2x - 8) - 15 = 0 \]
Expanding this, we get:
\[ x^2 - 2mx + 8m - 15 = 0 \]
Here, \( a = 1 \), \( b = -2m \), and \( c = 8m - 15 \).

### Step 2: Condition for Equal Roots
For the quadratic equation to have equal roots, the discriminant must be zero:
\[ D = b^2 - 4ac = 0 \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ (-2m)^2 - 4(1)(8m - 15) = 0 \]
This simplifies to:
\[ 4m^2 - 32m + 60 = 0 \]

### Step 3: Solve the Quadratic Equation
Dividing the entire equation by 4:
\[ m^2 - 8m + 15 = 0 \]
Factoring the quadratic:
\[ (m - 3)(m - 5) = 0 \]
Thus, we find:
\[ m = 3 \quad \text{or} \quad m = 5 \]

### Step 4: Condition for Both Roots Positive
For both roots to be positive, we need to ensure that the discriminant is greater than zero and the roots themselves are positive.

1. **Discriminant Condition**:
\[ D = b^2 - 4ac > 0 \]
Using the same values:
\[ 4m^2 - 32m + 60 > 0 \]
This can be factored as:
\[ (m - 3)(m - 5) > 0 \]

2. **Finding Intervals**:
The critical points are \( m = 3 \) and \( m = 5 \). We analyze the sign of the product in the intervals:
- For \( m < 3 \): both factors are negative, so the product is positive.
- For \( 3 < m < 5 \): one factor is negative and the other is positive, so the product is negative.
- For \( m > 5 \): both factors are positive, so the product is positive.

Thus, the solution for the discriminant condition is:
\[ m < 3 \quad \text{or} \quad m > 5 \]

3. **Ensuring Positive Roots**:
For the roots to be positive, we also need to ensure that \( c > 0 \):
\[ 8m - 15 > 0 \implies m > \frac{15}{8} = 1.875 \]

### Final Conditions
Combining the conditions:
- From the discriminant: \( m < 3 \) or \( m > 5 \)
- From the positivity of roots: \( m > 1.875 \)

Thus, the values of \( m \) for which the quadratic equation has:
1. Equal roots: \( m = 3 \) or \( m = 5 \)
2. Both roots positive: \( m \in (-\infty, 3) \cup (5, \infty) \)