If 4a is a perfect square, what is the next greater perfect square?

Answer 1

4a is a perfect square, which means that the square root (2√a) is an integer.

The next integer would be:
2√a + 1

And its square would be:
(2√a + 1)²

Using FOIL:

(2√a)² + 4√a + 1

This comes out to:
4a + 4√a + 1

For example, if a = 1, then 4a = 4,
and the next perfect square would be:
4(1) + 4√1 + 1 = 9

Or if a = 4, then 4a = 16. The next perfect square should be 25. Let's see:
4(4) + 4√4 + 1
= 16 + 8 + 1
= 25

And a final example, if a = 1/4, then 4a = 1. The next perfect square should be 4.
4(1/4) + 4√(1/4) + 1
= 1 + 4(1/2) + 1
= 1 + 2 + 1
= 4

The correct answer is:
4a + 4√a + 1

Answer 2

If 4a is a perfect square, then you could say that 4a=n^2 for some n. The next perfect square would be (n+1)^2.

(n+1)^2 = n^2 + 2n + 1
You need to notice that 4a being a perfect square implies that a is non-negative. Because of that sqrt(a) is real (actually would also be an integer, but real is fine for now). So n=2sqrt(a). That means the next largest perfect square is 4a+4sqrt(a)+1.

Answer 3

4a = x²
x = 2√a

(x+1)² = x² + 2x + 1
= x² + 4√a + 1
= 4a + 4√a + 1


The next perfect square is 4a+4√a+1

Answer 4

4a is a perfect square (you said), next perfect square greater than 4a is 4.0401a also.

Answer 5

It would have to be 9a. All we really need to know is that a is a perfect square, and the next perfect square after 4 is 9, so, yes, it would be 9a.

that's it! :)

Answer 6

4a+4sqrt{a}+1