Answer:
[tex]a=\dfrac{n^2-P^2n}{P^2-1}[/tex]
Step-by-step explanation:
Given equation:
[tex]P=\sqrt{\dfrac{n^2+a}{n+a}}[/tex]
Square both sides of the equation:
[tex]\implies P^2=\dfrac{n^2+a}{n+a}[/tex]
Multiply both sides by (n + a):
[tex]\implies P^2(n+a)=n^2+a[/tex]
Expand the parentheses:
[tex]\implies P^2n+P^2a=n^2+a[/tex]
Subtract a from both sides:
[tex]\implies P^2n+P^2a-a=n^2[/tex]
Subtract P²n from both sides:
[tex]\implies P^2a-a=n^2-P^2n[/tex]
Factor out a on the left side of the equation:
[tex]\implies a(P^2-1)=n^2-P^2n[/tex]
Divide both sides by (P² - 1):
[tex]\implies a=\dfrac{n^2-P^2n}{P^2-1}[/tex]