A square garden has an area of 64 square feet. If the side length is x feet, the relationship is x² = 64.
The algebra gives two mathematical solutions, but only one makes physical sense as a side length. Which choice gives BOTH algebraic solutions AND correctly identifies the physical side length?
AOnly x = 8 — that's the only solution to x² = 64
BOnly x = −8, because squaring a negative gives a positive
Cx = ±8 algebraically; the garden's side length is +8 ft
Dx = 32, because half of 64 is the side length
Explanation
Taking the square root of both sides gives x = ±√64 = ±8, because both (+8)² = 64 AND (−8)² = 64. In pure algebra you must report both roots. For the physical garden, length can't be negative, so only +8 ft applies in context — but the algebraic answer set is {−8, +8}.
Common mistakes: (A) Dropping the negative root entirely. (D) Dividing 64 by 2 instead of taking a square root.
Tip: when you see x² = N, always write x = ±√N first, then decide which roots fit the real-world context.