The complete set of values of a for which the roots of the equatin `x ^(2) -2 |a+1| x +1 =0` are real is given by :

Options

A. `(oo, -2] uu [0,oo)`
B. `(-oo, -1] uu [0,oo)`
C. `(-oo, -1]uu [1,oo)`
D. `(-oo,-2] uu[1,oo)`

Answer

To determine the complete set of values of \( a \) for which the roots of the equation

\[ x^2 - 2|a+1|x + 1 = 0 \]

are real, we need to analyze the discriminant of the quadratic equation. The roots of a quadratic equation are real if the discriminant \( D \) is greater than or equal to zero.

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

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

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

Substituting the coefficients we identified:

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

### Step 3: Simplify the discriminant
Calculating \( D \):

\[ D = 4|a+1|^2 - 4 \]

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

\[ 4|a+1|^2 - 4 \geq 0 \]

### Step 5: Factor out the common term
Dividing the entire inequality by 4:

\[ |a+1|^2 - 1 \geq 0 \]

### Step 6: Rewrite the inequality
This can be rewritten as:

\[ |a+1|^2 \geq 1 \]

### Step 7: Solve the absolute value inequality
Taking the square root of both sides, we have two cases:

1. \( |a+1| \geq 1 \)

This leads to two sub-cases:

- \( a + 1 \geq 1 \) which simplifies to \( a \geq 0 \)
- \( a + 1 \leq -1 \) which simplifies to \( a \leq -2 \)

### Step 8: Combine the results
From the above cases, we find that the values of \( a \) for which the roots are real are:

\[ a \leq -2 \quad \text{or} \quad a \geq 0 \]

### Final Answer
Thus, the complete set of values of \( a \) for which the roots of the equation are real is:

\[ (-\infty, -2] \cup [0, \infty) \]