Question

Fill in the blank to make the expression a perfect square:

Answer 1

Answer:

64

Step-by-step explanation:

So here you're just completing the square. the equation you gave is simply: $n^2+16n+c$ 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: $(x+b)^2$ and this binomial expands out to become: $x^2+2bx+b^2$

If we write the second term of the binomial as b/2 we get:

$(x+\frac{b}{2})^2=x^2+2(\frac{b}{2})x+(\frac{b}{2})^2$

which simplifies to:

$(x+\frac{b}{2})^2=x^2+bx+(\frac{b}{2})^2$

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: $(n+8)^2$ which expands out to: $n^2+2(8)(n)+8^2 = n^2+16n+64$