Question

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?: x^2 - 14x + c = 13

Answer 1

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.

1. The coefficient of \( x \) in the expression is \(-14\).
2. 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
\[
\boxed{49}
\]