The equation is given as:

$$23 - 7x = \sqrt{3k - 2x}$$

In the given equation, $k$ is a constant. The equation has exactly one real solution. We are tasked with finding the minimum possible value of $56k$.

Solution:

Square both sides of the equation to eliminate the square root:

$$(23 - 7x)^2 = 3k - 2x$$

Expanding the left-hand side:

$$529 - 322x + 49x^2 = 3k - 2x$$

Rearranging terms:

$$49x^2 - 320x + (529 - 3k) = 0$$

For the quadratic equation to have exactly one real solution, the discriminant ($b^2 - 4ac$) must equal zero:

$$b^2 - 4ac = 0$$

Here, $a = 49$, $b = -320$, and $c = 529 - 3k$. Substituting these values into the discriminant:

$$(-320)^2 - 4(49)(529 - 3k) = 0$$

Calculate each term:

$$102400 - 4(49)(529 - 3k) = 0$$ $$102400 - 103684 + 588k = 0$$ $$-1284 + 588k = 0$$

Solve for $k$:

$$588k = 1284$$ $$k = \frac{1284}{588} = \frac{107}{49}$$

Finally, calculate $56k$:

$$56k = 56 \cdot \frac{107}{49} = \frac{856}{7}$$

Conclusion: The minimum possible value of $56k$ is:

Answer: B) $\frac{856}{7}$