Find the values of the parameter a for which the roots of the quadratic equation `x^(2)+2(a-1)x+a+5=0` are
greater than 3.

Answer

To find the values of the parameter \( a \) for which the roots of the quadratic equation \( x^2 + 2(a-1)x + (a+5) = 0 \) are both greater than 3, we will follow these steps:

### Step 1: Identify the quadratic equation
The given quadratic equation is:
\[ f(x) = x^2 + 2(a-1)x + (a+5) \]

### Step 2: Determine the conditions for the roots to be greater than 3
For both roots of the quadratic equation to be greater than 3, we need to satisfy two conditions:
1. The quadratic must have real roots (discriminant \( D \geq 0 \)).
2. The value of the quadratic at \( x = 3 \) must be positive, i.e., \( f(3) > 0 \).

### Step 3: Calculate the discriminant
The discriminant \( D \) of the quadratic equation \( ax^2 + bx + c \) is given by:
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = 2(a-1) \), and \( c = a + 5 \). Thus,
\[ D = [2(a-1)]^2 - 4(1)(a+5) \]
Calculating this:
\[ D = 4(a-1)^2 - 4(a+5) \]
\[ D = 4[(a-1)^2 - (a+5)] \]
\[ D = 4[a^2 - 2a + 1 - a - 5] \]
\[ D = 4[a^2 - 3a - 4] \]
For the roots to be real, we need:
\[ a^2 - 3a - 4 \geq 0 \]

### Step 4: Solve the inequality
To solve \( a^2 - 3a - 4 \geq 0 \), we first find the roots of the equation \( a^2 - 3a - 4 = 0 \) using the quadratic formula:
\[ a = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-4)}}{2(1)} = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm 5}{2} \]
This gives us:
\[ a = 4 \quad \text{and} \quad a = -1 \]
The inequality \( a^2 - 3a - 4 \geq 0 \) holds for:
\[ a \leq -1 \quad \text{or} \quad a \geq 4 \]

### Step 5: Calculate \( f(3) \)
Next, we need to check the condition \( f(3) > 0 \):
\[ f(3) = 3^2 + 2(a-1)(3) + (a+5) \]
Calculating this:
\[ f(3) = 9 + 6(a-1) + (a+5) \]
\[ f(3) = 9 + 6a - 6 + a + 5 \]
\[ f(3) = 7a + 8 \]
We need:
\[ 7a + 8 > 0 \implies a > -\frac{8}{7} \]

### Step 6: Combine the conditions
Now we have two conditions:
1. \( a \leq -1 \) or \( a \geq 4 \)
2. \( a > -\frac{8}{7} \)

### Step 7: Analyze the intervals
- For \( a \leq -1 \): This interval does not satisfy \( a > -\frac{8}{7} \).
- For \( a \geq 4 \): This interval satisfies \( a > -\frac{8}{7} \).

### Conclusion
The values of \( a \) for which both roots of the quadratic equation are greater than 3 are:
\[ \boxed{a \geq 4} \]