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 will follow these steps:

### Step 1: Identify the coefficients
From the given quadratic equation, we can identify the coefficients:
- \( a = 1 \)
- \( b = 2(a-1) \)
- \( c = a + 5 \)

### Step 2: Write the condition for real and distinct roots
For the roots of a quadratic equation to be real and distinct, the discriminant (\( D \)) must be greater than 0. The discriminant is given by:

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

### Step 3: Substitute the coefficients into the discriminant
Substituting \( b \) and \( c \) into the discriminant formula:

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

### Step 4: Simplify the discriminant
Now we simplify the expression:

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

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

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

Distributing the 4:

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

Combining like terms:

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

### Step 5: Set the discriminant greater than zero
To find the values of \( a \) for which the roots are real and distinct, we set the discriminant greater than zero:

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

### Step 6: Factor the quadratic inequality
We can factor out the 4 from the inequality:

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

Dividing both sides by 4 (since 4 is positive, the inequality sign remains the same):

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

### Step 7: Factor the quadratic expression
Now we will factor the quadratic expression:

\[ a^2 - 3a - 4 = (a - 4)(a + 1) \]

### Step 8: Solve the inequality
We need to solve the inequality:

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

### Step 9: Determine the critical points
The critical points are \( a = 4 \) and \( a = -1 \). We will test the intervals determined by these points:

1. \( (-\infty, -1) \)
2. \( (-1, 4) \)
3. \( (4, \infty) \)

### Step 10: Test the intervals
- For \( a < -1 \) (e.g., \( a = -2 \)):
\((a - 4)(a + 1) = (-2 - 4)(-2 + 1) = (-6)(-1) > 0\) (True)

- For \( -1 < a < 4 \) (e.g., \( a = 0 \)):
\((a - 4)(a + 1) = (0 - 4)(0 + 1) = (-4)(1) < 0\) (False)

- For \( a > 4 \) (e.g., \( a = 5 \)):
\((a - 4)(a + 1) = (5 - 4)(5 + 1) = (1)(6) > 0\) (True)

### Step 11: Write the solution
The solution to the inequality is:

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