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
equal

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 equal, we need to use the condition that the discriminant of the quadratic equation is equal to zero.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \).
Here, we have:
- \( a = 1 \)
- \( b = 2(a - 1) \)
- \( c = a + 5 \)

2. **Write the discriminant**:
The discriminant \( D \) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
For our equation, substituting the values of \( a \), \( b \), and \( c \):
\[ D = [2(a - 1)]^2 - 4(1)(a + 5) \]

3. **Simplify the discriminant**:
Calculate \( D \):
\[ D = 4(a - 1)^2 - 4(a + 5) \]
Expanding \( D \):
\[ D = 4[(a - 1)^2 - (a + 5)] \]
\[ = 4[a^2 - 2a + 1 - a - 5] \]
\[ = 4[a^2 - 3a - 4] \]

4. **Set the discriminant to zero**:
For the roots to be equal, we set the discriminant \( D \) to zero:
\[ 4(a^2 - 3a - 4) = 0 \]
Dividing both sides by 4:
\[ a^2 - 3a - 4 = 0 \]

5. **Factor the quadratic equation**:
Now, we need to factor \( a^2 - 3a - 4 \):
\[ (a - 4)(a + 1) = 0 \]

6. **Find the values of \( a \)**:
Setting each factor to zero gives us:
\[ a - 4 = 0 \quad \Rightarrow \quad a = 4 \]
\[ a + 1 = 0 \quad \Rightarrow \quad a = -1 \]

### Final Answer:
The values of the parameter \( a \) for which the roots of the quadratic equation are equal are:
\[ a = 4 \quad \text{and} \quad a = -1 \]