< b="" />Solution:
Given equation: x^2 + y^2 + x + c = 0

To find the radius of the circle passing through the origin, we need to find the center of the circle first.

< b="" />Step 1: Completing the square
To simplify the equation, we can complete the square by adding and subtracting a constant term.
x^2 + y^2 + x + c = 0

Rearrange the equation:
x^2 + x + y^2 + c = 0

To complete the square, we need to add and subtract (1/2)^2 = 1/4 inside the parentheses:
x^2 + x + 1/4 + y^2 + c - 1/4 = 0

Rewrite the equation:
(x + 1/2)^2 + y^2 + c - 1/4 = 0

< b="" />Step 2: Identify the center and radius
Comparing the equation to the standard form of a circle equation:
(x - h)^2 + (y - k)^2 = r^2

We can see that the center of the circle is (-1/2, 0) and the square of the radius is c - 1/4.

< b="" />Step 3: Find the radius
To find the radius, we take the square root of the square of the radius:
r = √(c - 1/4)

Since we want to find the radius passing through the origin, we substitute (0, 0) into the equation to get:
r = √(c - 1/4)

< b="" />Step 4: Determine the correct option
From the given options, we need to find the value of c that gives us the radius of 1/2.

Substitute r = 1/2 into the equation:
1/2 = √(c - 1/4)

Square both sides to eliminate the square root:
1/4 = c - 1/4

Simplify the equation:
c = 1/4 + 1/4

c = 1/2

Therefore, the correct option is < b="" />B) 1/2.