To determine the value of \( c \) that would make the left-hand side \( x^2 - 14x + c \) a perfect square trinomial, we can use the method of completing the square.
A perfect square trinomial can be expressed in the form \( (x - k)^2 = x^2 - 2kx + k^2 \). Thus, we need to find \( c \) such that the expression \( x^2 - 14x + c \) matches this form.
- The coefficient of \( x \) in the expression is \(-14\).
- To complete the square, we take half of this coefficient, square it, and add it to the expression.
First, calculate half of \(-14\): \[ \frac{-14}{2} = -7 \]
Next, square this result: \[ (-7)^2 = 49 \]
Thus, \( c \) should be \( 49 \) in order for \( x^2 - 14x + c \) to be a perfect square trinomial. The complete trinomial then would be: \[ x^2 - 14x + 49 = (x - 7)^2 \]
So, the value of \( c \) that makes the left-hand side a perfect square trinomial is \[ \boxed{49} \]