`sqrt( -1 -sqrt(1-sqrt(-1-"...."oo)))` is equal to

Options

A. 1
B. `-1`
C. `omega^(2)`
D. `-omega`

Answer

To solve the expression \( \sqrt{-1 - \sqrt{1 - \sqrt{-1 - \ldots}}} \), we can define the entire expression as \( Y \). Thus, we have:

\[ Y = \sqrt{-1 - \sqrt{1 - Y}} \]

### Step 1: Square both sides
To eliminate the square root, we square both sides:

\[ Y^2 = -1 - \sqrt{1 - Y} \]

### Step 2: Isolate the remaining square root
Next, we isolate the square root term:

\[ \sqrt{1 - Y} = -1 - Y^2 \]

### Step 3: Square both sides again
Now, we square both sides again to eliminate the square root:

\[ 1 - Y = (-1 - Y^2)^2 \]

### Step 4: Expand the right side
Expanding the right side gives:

\[ 1 - Y = 1 + 2Y^2 + Y^4 \]

### Step 5: Rearrange the equation
Rearranging the equation leads to:

\[ Y^4 + 2Y^2 + Y = 0 \]

### Step 6: Factor the equation
Factoring out \( Y \):

\[ Y(Y^3 + 2Y + 1) = 0 \]

This gives us one solution \( Y = 0 \). We now need to solve the cubic equation \( Y^3 + 2Y + 1 = 0 \).

### Step 7: Use the cubic formula or synthetic division
To find the roots of the cubic equation, we can use the cubic formula or synthetic division. However, for simplicity, we will use the Rational Root Theorem to test possible rational roots. Testing \( Y = -1 \):

\[ (-1)^3 + 2(-1) + 1 = -1 - 2 + 1 = -2 \quad (\text{not a root}) \]

Testing \( Y = -1 \) again:

\[ (-1)^3 + 2(-1) + 1 = -1 + 2 + 1 = 2 \quad (\text{not a root}) \]

### Step 8: Find the discriminant
To find the nature of the roots, we can calculate the discriminant of the cubic polynomial. The discriminant \( D \) for a cubic equation \( ax^3 + bx^2 + cx + d = 0 \) is given by:

\[ D = 18abcd - 4b^3d + b^2c^2 - 4ac^2 - 27a^2d^2 \]

For our equation \( Y^3 + 0Y^2 + 2Y + 1 = 0 \), we have \( a = 1, b = 0, c = 2, d = 1 \).

Calculating the discriminant:

\[ D = 18(1)(0)(2)(1) - 4(0)^3(1) + (0)^2(2)^2 - 4(1)(2)^2 - 27(1)^2(1)^2 \]
\[ D = 0 - 0 + 0 - 16 - 27 = -43 \]

Since the discriminant is negative, this indicates that the cubic equation has one real root and two complex conjugate roots.

### Step 9: Use numerical methods or graphing
To find the approximate value of the real root, we can use numerical methods or graphing techniques. The real root can be approximated to be around \( -1.325 \).

### Conclusion
The value of the original expression \( \sqrt{-1 - \sqrt{1 - \sqrt{-1 - \ldots}}} \) converges to a complex number, which can be expressed in terms of \( \omega \) and \( \omega^2 \) as described in the video.