Given information:
- The three vertices of the equilateral triangle lie on the parabola y = x^2.
- One side of the triangle has a slope of 2.

Let's solve this problem step by step:

Step 1: Finding the slope of the side of the triangle
Since one side of the triangle has a slope of 2, we need to find the equation of the line passing through two points on the parabola that have a slope of 2. Let's consider two points (x1, y1) and (x2, y2) on the parabola.

Since the parabola equation is y = x^2, we can substitute the values of x1 and x2 into the equation to get y1 and y2.

y1 = x1^2
y2 = x2^2

Now, we can find the slope between these two points:

slope = (y2 - y1) / (x2 - x1)
= (x2^2 - x1^2) / (x2 - x1)
= (x2 + x1)(x2 - x1) / (x2 - x1)
= x2 + x1

Since the slope is given as 2, we have:

2 = x2 + x1

Step 2: Finding the coordinates of the vertices of the equilateral triangle
Since the three vertices of the equilateral triangle lie on the parabola, we can substitute the value of x into the equation of the parabola to find the corresponding y-coordinate.

Let's assume the coordinates of the vertices of the equilateral triangle are (x1, y1), (x2, y2), and (x3, y3).

Substituting x1, x2, and x3 into the equation of the parabola, we get:

y1 = x1^2
y2 = x2^2
y3 = x3^2

Step 3: Applying the condition of an equilateral triangle
Since the triangle is equilateral, the distance between any two vertices should be the same.

Let's consider two vertices (x1, y1) and (x2, y2).

Using the distance formula, the distance between these two vertices is:

d1 = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Since the triangle is equilateral, the distance between any two vertices should be the same. Therefore, we also have:

d2 = sqrt((x3 - x1)^2 + (y3 - y1)^2) = d1
d3 = sqrt((x3 - x2)^2 + (y3 - y2)^2) = d1

Step 4: Solving the equations
Using the equations from Step 2 and Step 3, we can solve for x1, x2, and x3.

From Step 2:
y1 = x1^2
y2 = x2^2
y3 = x3^2

From Step 3:
d1 = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d2 = sqrt((x3 - x1)^2 + (y3 - y1)^2)
d3 = sqrt((x3 - x2