The distance between the points (0, −5) and (x, 0) is 13 units. Find the possible value(s) of x.

Introduction / Context:
The two-dimensional distance formula connects coordinate differences to the straight-line distance between points. Here, one point lies on the y-axis and the other on the x-axis, which simplifies computation because one coordinate in each point is zero.

Given Data / Assumptions:

• Point A = (0, −5).
• Point B = (x, 0).
• Distance AB = 13 units.

Concept / Approach:
Use distance formula: For A(x1, y1) and B(x2, y2), distance d satisfies d^2 = (x2 − x1)^2 + (y2 − y1)^2. Substitute the known values, square the distance, and solve for x, remembering that squaring leads to two symmetric solutions (±) when isolating x.

Step-by-Step Solution:

Verification / Alternative check:
Check x = 12: distance^2 = 12^2 + 5^2 = 144 + 25 = 169 ⇒ distance = 13. Similarly for x = −12 because x^2 is the same; both satisfy the condition.

Why Other Options Are Wrong:

• 10, ±10, 12, 13: These do not satisfy x^2 = 144 when substituted into the formula; only ±12 works.

Common Pitfalls:

• Dropping the negative solution and reporting only +12.
• Using 13 directly as x rather than applying the distance relation.

Final Answer:
± 12