Square (algebra)

The square of a number is the result of the multiplication of this number by itself. The opposite operation of the square is the square root.

Examples:

5 ² = 25
1 ² = 1
10 ² = 100
$\ sqrt {100} = 10$

General information on the square

When one multiplies a number squared, one multiplies it by itself. Thus, the forms 12 ² and 12 X 12 are equivalent. Nevertheless one prefers the form 12 ² as much as possible for his clearness and his concision. A square is always positive for any real number .

Example: 12 ² = (- 12) ² = 12 X 12 = -12 X (- 12) = 144

Caution! - (12 ²) and (- 12) ² is two different numbers . The first is worth -144 (one multiplies 12 by 12 then by -1) and second 144 (it - is included in the bracket).

The square root

As one can raise a number with the square , one can also make the opposite operation. That is called the square root of a number. That is to say has a positive number , the square root of has is written: $\ sqrt a$ It is important to specify that has must be positive. Indeed to write $\ sqrt {- has}$ for example would amount saying that ² < 0 has what is not possible in the whole of the real . On the other hand, as it is possible to write - (12 ²) it is completely possible to write $- \ sqrt a$

To solve equation X ² has in the whole of realities

First case: < 0 have

When has is strictly lower than 0, in other words, X ² is negative. However in the whole of realities, the square of a number is never negative. Thus: $S = \ empty$

Second case: 0 have

When has is worth 0, only one solution is possible: 0 (since zero do not have a sign). Thus: $S = \ left \ {0 \ right \}$

Third case: > 0 have

We saw in the preceding part that 12 ² = (- 12) ² = 144. One can réappliquer this assertion with equation X ² = A. Ici the equation thus has two solutions: $S = \ left \ {- \ sqrt has; \ sqrt has \ right \}$

Note:: to solve $\ sqrt X = a$

If has is strictly negative, the equation does not have a solution. Thus: $S = \ empty$

On the other hand if $a \ Ge 0$ then to find X amounts multiplying has by itself, i.e. ² has.

Notes and references of the article

Primary source of this article: course of mathematics level 3ème/2nde

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