Answer:
64
Step-by-step explanation:
So here you're just completing the square. the equation you gave is simply: [tex]n^2+16n+c[/tex] where c is the unknown value we're solving for. Whenever you complete the square, you add (b/2)^2
The reason for this, is because whenever you write a binomial as a perfect square it's in the form: [tex](x+b)^2[/tex] and this binomial expands out to become: [tex]x^2+2bx+b^2[/tex]
If we write the second term of the binomial as b/2 we get:
[tex](x+\frac{b}{2})^2=x^2+2(\frac{b}{2})x+(\frac{b}{2})^2[/tex]
which simplifies to:
[tex](x+\frac{b}{2})^2=x^2+bx+(\frac{b}{2})^2[/tex]
and as you can see the last term is (b/2)^2, which is why we need to add that part for it to be a perfect square.
So we would need to add (16/2)^2 = 8^2 = 64
This way, we can express it as a perfect square binomial: [tex](n+8)^2[/tex] which expands out to: [tex]n^2+2(8)(n)+8^2 = n^2+16n+64[/tex]
