To determine the correct equation for [tex]\( k \)[/tex] when given the potential energy formula [tex]\( P = \frac{1}{2} k x^2 \)[/tex], we need to solve for [tex]\( k \)[/tex].
Let's start with the given equation:
[tex]\[ P = \frac{1}{2} k x^2 \][/tex]
To isolate [tex]\( k \)[/tex], follow these steps:
1. Multiply both sides by 2 to eliminate the [tex]\(\frac{1}{2}\)[/tex] on the right side:
[tex]\[ 2P = k x^2 \][/tex]
2. Now, divide both sides by [tex]\( x^2 \)[/tex] to solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{2P}{x^2} \][/tex]
This manipulation shows that the correct equation for [tex]\( k \)[/tex] is:
[tex]\[ k = \frac{2P}{x^2} \][/tex]
Thus, the correct option to use is:
[tex]\[ \boxed{k = \frac{2P}{x^2}} \][/tex]
