2070.71 – Next Perfect Square

If xx is a perfect square (i.e., the square of some integer), then the next perfect square larger than xx is:

  1. x+1x + 1
  2. x2+1x^2 + 1
  3. x2+2x+1x^2 + 2x + 1
  4. x2+xx^2 + x
  5. x+2x+1x + 2\sqrt{x} + 1
Solution

Let x=n2x = n^2. Then x=n\sqrt{x} = n. The next perfect square after xx will be the square of n+1n + 1. So we want (n+1)2(n + 1)^2 in terms of xx.

(n+1)2=n2+2n+1=x+2x+1(n + 1)^2 = n^2 + 2n + 1 = x + 2\sqrt{x} + 1. The answer is (e).

Check: Let x = 49; then nn = 7. So n+1n + 1 = 8, and (n+1)2(n + 1)^2 = 64.

Is 64 the next perfect square after 49? Yes.

Does x+2x+1x + 2\sqrt{x}+ 1 = 64? Yes, 49 + 2 · 7 + 1 = 49 + 14 + 1 = 64.

Ah, what more could anyone ask?