Which of the following is correct for the quadratic equation `x^(2)+2(a-1)x+a+5=0` ?

Options

A. the equation has positive roots, if `a in (-5, -1)`
B. the equation has roots of opposite sign, if `a in(-oo, -5)`
C. the equation has negative roots, if `a in [4, oo)`
D. none of these

Answer

To solve the quadratic equation \( x^2 + 2(a-1)x + (a+5) = 0 \) and determine the conditions for its roots, we will analyze the discriminant and the nature of the roots based on the value of \( a \).

### Step 1: Identify the coefficients
The given quadratic equation can be rewritten in the standard form \( ax^2 + bx + c = 0 \) where:
- \( a = 1 \)
- \( b = 2(a-1) = 2a - 2 \)
- \( c = a + 5 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (2a - 2)^2 - 4(1)(a + 5) \]
Expanding this:
\[ D = (4a^2 - 8a + 4) - (4a + 20) \]
\[ D = 4a^2 - 8a + 4 - 4a - 20 \]
\[ D = 4a^2 - 12a - 16 \]

### Step 3: Set the discriminant greater than or equal to zero
For the roots to be real, the discriminant must be non-negative:
\[ 4a^2 - 12a - 16 \geq 0 \]
Dividing the entire inequality by 4:
\[ a^2 - 3a - 4 \geq 0 \]

### Step 4: Factor the quadratic expression
Factoring the quadratic:
\[ (a - 4)(a + 1) \geq 0 \]

### Step 5: Determine the intervals
To find the intervals where this inequality holds, we find the roots:
- \( a - 4 = 0 \) gives \( a = 4 \)
- \( a + 1 = 0 \) gives \( a = -1 \)

Now, we test the intervals:
1. \( (-\infty, -1) \)
2. \( (-1, 4) \)
3. \( (4, \infty) \)

Testing these intervals:
- For \( a < -1 \): Choose \( a = -2 \) → \( (-2 - 4)(-2 + 1) = (-6)(-1) > 0 \) (True)
- For \( -1 < a < 4 \): Choose \( a = 0 \) → \( (0 - 4)(0 + 1) = (-4)(1) < 0 \) (False)
- For \( a > 4 \): Choose \( a = 5 \) → \( (5 - 4)(5 + 1) = (1)(6) > 0 \) (True)

Thus, the solution to the inequality is:
\[ a \in (-\infty, -1] \cup [4, \infty) \]

### Step 6: Analyze the nature of the roots
1. **Positive Roots**: For both roots to be positive, we need:
- \( \alpha + \beta > 0 \) (which gives \( -b/a > 0 \))
- \( \alpha \beta > 0 \) (which gives \( c/a > 0 \))

From \( \alpha + \beta = -2a + 2 > 0 \) → \( a < 1 \)
From \( \alpha \beta = a + 5 > 0 \) → \( a > -5 \)

Thus, for positive roots:
\[ a \in (-5, 1) \]

2. **Opposite Sign Roots**: For the roots to have opposite signs:
- One root positive and the other negative implies:
\[ \alpha + \beta > 0 \quad \text{and} \quad \alpha \beta < 0 \]
This leads to:
\[ a < -5 \]

3. **Negative Roots**: For both roots to be negative:
- \( \alpha + \beta < 0 \) and \( \alpha \beta > 0 \):
\[ -2a + 2 < 0 \quad \Rightarrow \quad a > 1 \]
\[ a + 5 > 0 \quad \Rightarrow \quad a > -5 \]
Thus, for negative roots:
\[ a \in (1, \infty) \]

### Conclusion
Based on the analysis:
- The equation has positive roots if \( a \in (-5, 1) \).
- The equation has roots of opposite signs if \( a < -5 \).
- The equation has negative roots if \( a > 1 \).