# Question #bc307

##### 3 Answers

#### Explanation:

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"

**Historical note**

Isaac Newton, one of the founders of physics and calculus, referred to solution to

**Personal note**

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:

#### Explanation:

In the first equation

and you'll get zero:

and

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.

#### Explanation:

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 *two zeros* at

and

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**.

So, if

Moreover, this expression will never be equal to zero, because that would imply that

something that cannot happen for real numbers.