Final answer:
In comparing the two expressions (2x^2) and (x + y)^2 for x < y, the expression (x + y)^2 is greater because adding y (which is larger than x) before squaring results in a larger number than just squaring x and multiplying by 2.
Step-by-step explanation:
The student is asking to compare two algebraic expressions, (2x^2) and (x + y)^2, given that x and y are the numbers of students in two classrooms with x being less than y (x < y). To compare these expressions, we can analyze their components.
- For the first expression, (2x^2), since x is a positive number representing the count of students, this expression will also yield a positive result when x is squared and then multiplied by 2.
- The second expression, (x + y)^2, represents the square of the sum of x and y. Since y is larger than x, the sum x + y is greater than x alone. Therefore, when squared, (x + y)^2 will yield a larger number than x squared (x^2).
Thus, we can conclude that (x + y)^2 is greater than (2x^2) because the addition of y (which is greater than x) before squaring will produce a larger result.
We can also illustrate this with a simple example. Suppose x = 2 and y = 3, which satisfies the condition x < y.
- For (2x^2), calculating gives 2*(2^2) = 2*4 = 8.
- For (x + y)^2, calculating gives (2 + 3)^2 = 5^2 = 25.
Clearly, 25 > 8, so (x + y)^2 > (2x^2).