You notice that the following problem cannot be factored so you solve it by completing the square. What value of c would make the left hand side of this equation a perfect square trinomial?

To determine the value of \( c \) that would make the expression \( x^2 - 14x + c \) a perfect square trinomial, we can use the method of completing the square.

A perfect square trinomial can be expressed in the form \( (x - a)^2 = x^2 - 2ax + a^2 \).

In our case, we have:

\[
x^2 - 14x + c
\]

Here, the coefficient of \( x \) is \(-14\). To find the value of \( c \) that makes this a perfect square, we take half of \(-14\) and square it:

1. Calculate half of \(-14\):
\[
-\frac{14}{2} = -7
\]

2. Now square that value:
\[
(-7)^2 = 49
\]

Thus, if \( c = 49 \), the expression \( x^2 - 14x + c \) becomes a perfect square trinomial:

\[
x^2 - 14x + 49 = (x - 7)^2
\]

Now, we have the equation:

\[
x^2 - 14x + 49 = 13
\]

Therefore, the value of \( c \) that would make the left-hand side a perfect square trinomial is:

\[
\boxed{49}
\]