The is no Real number whose square is
In many textbooks, this kind of equation is said to have "no solution until the author gets around to introducing
(The square of a positive is positive and the square of a negative is positive.)
We have to invent new kinds of numbers to get a zero for
Don't think of it as a big deal. For many centuries, the only "numbers" were the counting numbers. There were these things called "ratios of numbers" but for a long time they were not thought of as "numbers".
We need them to solve equations like
Similarly, ratios (fractions) of whole numbers will not allow us to solve
In a similar way, solving
Due to objections to the introduction of this new kind of "number" they were called "Imaginary Numbers"
Isaac Newton, one of the founders of physics and calculus, referred to solution to
I still have issues 50 years later from learning (in 2nd grade) that
When I learned about fractions the next year I was angry! I thought (and still think) I should have been taught
Have a look if it is clear:
In the first equation
and you'll get zero:
But in the second expression you cannot find two real numbers that makes it equal to zero.
If you try a positive number you'll square it and add to
If you try a negative number the square will change it into a positive number and, together with
It should be better to say that your second expression doesn't have REAL zeros.
Here's why that's so.
Your first expression
#x^2 - 25#
can actually be factored as the difference of two squares
#color(blue)(a^2 - b^2 = (a-b)(a+b))#
This means that you can write
This expression will be equal to zero when
On the other hand, your second expression doesn't have any zeros that are real numbers.
That happens because the square of any real number, positive or negative, is always positive.
Moreover, this expression will never be equal to zero, because that would imply that
something that cannot happen for real numbers.