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
real and distinct

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 real and distinct, we need to analyze the discriminant of the quadratic equation. The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are real and distinct if the discriminant \( D \) is greater than zero.

### Step 1: Identify coefficients
In our equation, we can identify:
- \( a = 1 \)
- \( b = 2(a-1) \)
- \( c = a + 5 \)

### Step 2: Write the discriminant
The discriminant \( D \) is given by the formula:

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

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = [2(a-1)]^2 - 4(1)(a+5) \]

### Step 3: Simplify the discriminant
Now, let's simplify the discriminant:

\[ D = 4(a-1)^2 - 4(a+5) \]

Expanding \( 4(a-1)^2 \):

\[ D = 4(a^2 - 2a + 1) - 4(a + 5) \]

Distributing the \( 4 \):

\[ D = 4a^2 - 8a + 4 - 4a - 20 \]

Combining like terms:

\[ D = 4a^2 - 12a - 16 \]

### Step 4: Set the discriminant greater than zero
For the roots to be real and distinct, we need:

\[ 4a^2 - 12a - 16 > 0 \]

### Step 5: Factor the quadratic expression
To solve the inequality, we can factor the quadratic expression:

First, divide the entire inequality by 4:

\[ a^2 - 3a - 4 > 0 \]

Now, factor the quadratic:

\[ (a - 4)(a + 1) > 0 \]

### Step 6: Find the critical points
The critical points from the factors are:

\[ a - 4 = 0 \Rightarrow a = 4 \]
\[ a + 1 = 0 \Rightarrow a = -1 \]

### Step 7: Test intervals on the number line
We will test the intervals determined by the critical points \( -1 \) and \( 4 \):

1. **Interval \( (-\infty, -1) \)**: Choose \( a = -2 \)
\[ (-2 - 4)(-2 + 1) = (-6)(-1) = 6 > 0 \quad \text{(True)} \]

2. **Interval \( (-1, 4) \)**: Choose \( a = 0 \)
\[ (0 - 4)(0 + 1) = (-4)(1) = -4 < 0 \quad \text{(False)} \]

3. **Interval \( (4, \infty) \)**: Choose \( a = 5 \)
\[ (5 - 4)(5 + 1) = (1)(6) = 6 > 0 \quad \text{(True)} \]

### Step 8: Conclusion
The solution to the inequality \( (a - 4)(a + 1) > 0 \) is:

\[ a \in (-\infty, -1) \cup (4, \infty) \]