Is 50 a perfect square?

3 Answers
Apr 12, 2015

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

Examples of perfect squares are:
http://images.tutorvista.com/cms/images/67/perfect-squares-chart11.JPG

Apr 12, 2015

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.

Jul 19, 2015

Answer:

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?