Question

Is 50 a perfect square?

Answer 1

50 is not a perfect square.
It does not have an exact square root.

Examples of perfect squares are:

Answer 2

An easy way you could find perfect squares is to memorize the first two, then add 2 to the differences. For example:

1, 4, 9, 16, 25, and 36 are the perfect squares up to `6^2`.

Now look at the differences.

`4 - 1 = 3`
`9 - 4 = 5`
`16 - 9 = 7`
`25 - 16 = 9`
`36 - 25 = 11`

See a pattern?

So, if you know that `24^2` is `576` and `25^2` is `625`, then `(625 - 576 + 2) + 625 = 26^2 = 676`

That is, simply take the difference of two consecutive squares, add `2`, then add it to the higher perfect square.

Answer 3

Here's an idea rather than an authoritative answer.

It may depend on the context. Normally "No", but possibly "Yes".

Explanation:

`50` is not the perfect square of an integer or rational number. This is what we normally mean by "a perfect square".

It is a square of an irrational, algebraic, real number, namely `5sqrt(2)`, therefore you could call it a perfect square in the context of the algebraic numbers.

For example, if you were asked to factor the polynomial `5x^2-1` you can usefully recognise this as a difference of squares:

`5x^2-1 = (sqrt(5)x)^2-1^2 = (sqrt(5)x - 1)(sqrt(5)x + 1)`

If recognising `5x^2` as a square means that we consider `5` as a perfect square being `(sqrt(5))^2`, then perhaps that's useful.

Another example:

We know that `x^2+2x+1 = (x+1)^2` is a perfect square trinomial.

What about `5x^2+10x+5`?

It is still the square of a binomial:

`5x^2+10x+5 = (sqrt(5)x + sqrt(5))^2`

In the context of polynomials, should we reserve the term 'perfect square' for polynomials with rational coefficients?