Let `lambda` be real. If. The origin and the complex roots of `2z^(2)+2z+lambda=0` form the three vertices b of an equilateral triangle, then `(1)/(lambda)` is equal to _________

Answer

To solve the problem, we need to find the value of \( \frac{1}{\lambda} \) given that the origin and the complex roots of the equation \( 2z^2 + 2z + \lambda = 0 \) form the vertices of an equilateral triangle.

### Step-by-Step Solution:

1. **Identify the roots of the quadratic equation**:
The roots of the equation \( 2z^2 + 2z + \lambda = 0 \) can be found using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = 2 \), and \( c = \lambda \).
Thus, the roots are:
\[ z = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 2 \cdot \lambda}}{2 \cdot 2} = \frac{-2 \pm \sqrt{4 - 8\lambda}}{4} = \frac{-1 \pm \sqrt{1 - 2\lambda}}{2} \]
Let the roots be \( z_1 = \frac{-1 + \sqrt{1 - 2\lambda}}{2} \) and \( z_2 = \frac{-1 - \sqrt{1 - 2\lambda}}{2} \).

2. **Sum and product of the roots**:
From Vieta's formulas:
- The sum of the roots \( z_1 + z_2 = -\frac{b}{a} = -\frac{2}{2} = -1 \).
- The product of the roots \( z_1 z_2 = \frac{c}{a} = \frac{\lambda}{2} \).

3. **Condition for equilateral triangle**:
The vertices of the equilateral triangle are the origin (0) and the roots \( z_1 \) and \( z_2 \). For three points to form an equilateral triangle, the following condition must hold:
\[ |z_1|^2 + |z_2|^2 = |z_1 + z_2|^2 + 3|z_1 z_2| \]
Since \( z_1 + z_2 = -1 \), we have:
\[ |z_1 + z_2|^2 = 1 \]

4. **Calculate \( |z_1|^2 + |z_2|^2 \)**:
Using the identity \( |z_1|^2 + |z_2|^2 = (z_1 + z_2)^2 - 2z_1 z_2 \):
\[ |z_1|^2 + |z_2|^2 = (-1)^2 - 2 \cdot \frac{\lambda}{2} = 1 - \lambda \]

5. **Set up the equation**:
Substitute into the condition for the equilateral triangle:
\[ 1 - \lambda = 1 + 3 \cdot \frac{\lambda}{2} \]
Simplifying gives:
\[ 1 - \lambda = 1 + \frac{3\lambda}{2} \]
Rearranging:
\[ -\lambda - \frac{3\lambda}{2} = 0 \]
\[ -\frac{5\lambda}{2} = 0 \implies \lambda = \frac{2}{3} \]

6. **Find \( \frac{1}{\lambda} \)**:
Now, we need to find \( \frac{1}{\lambda} \):
\[ \frac{1}{\lambda} = \frac{1}{\frac{2}{3}} = \frac{3}{2} \]

### Final Answer:
\[ \frac{1}{\lambda} = \frac{3}{2} \]