Equations of the form x² = a and practical applications

Solving x² = a

There are three possible cases, depending on the sign of a:

Value of a	Solutions to x² = a
a > 0	x = √a or x = -√a
a = 0	x = 0 (unique solution)
a < 0	no solutions

Example: x² = 25 gives x = 5 or x = -5. The set of solutions is S = {-5; 5}.

Application: area of a square

If a square has side length c, its area is A = c². This formula is a direct application of the square function, where c must be greater than 0 (a length cannot be negative).

Example: a square with a side length of 7 cm has an area of 7² = 49 cm². Conversely, if the area is 81 cm², the side length is √81 = 9 cm; only the positive solution is retained, as it represents a length.

Application: free fall

In physics, the distance travelled during free fall is proportional to the square of the elapsed time: d = k × t². This example shows that the square function is fundamental well beyond the realm of pure mathematics.

Pitfall to avoid

In a physical or geometric context, we generally retain only the positive solution to x² = a, as x represents a length or a duration. However, in pure mathematics, without any specific context, both solutions must always be given.