Visual models of identities

The square has side \(x+y\), so its area is \((x+y)^2\). Its partition contains \(x^2\), \(y^2\), and two \(xy\) rectangles, giving \(x^2+2xy+y^2\). Exam tip: count both equal \(xy\) regions.

Partitioning a square of side \((a+b)\) gives one \(a^2\) square, one \(b^2\) square, and two rectangles of area \(ab\). Hence the total is \(a^2+2ab+b^2\). Exam tip: count both \(ab\) rectangles.

In the subtraction model, the (b^2) corner is removed twice. It is added back once for the correct area.

In the area model for \((x+2)^2\), there are two middle rectangles. Each has side lengths \(x\) and \(2\), so each area is \(2x\). Therefore, their total area is \(2x+2x=4x\). \(2x\) is the area of only one rectangle. Exam tip: in \((a+b)^2\), the two middle regions together have area \(2ab\).

The grid has total length \(x+3\) and total width \(x+2\). Therefore, its total area is \((x+3)(x+2)=x^2+2x+3x+6=x^2+5x+6\). In option B, the coefficient of the middle term and the constant term are incorrectly interchanged. Exam tip: Add the areas of all four smaller rectangles in such a grid.

Here, x² - 9 = x² - 3². The term x² represents the area of a large square of side x, and 3² represents the area of a smaller square of side 3. Thus, removing the smaller square from the larger one leaves an area of x² - 9. A square of side x-3 has area (x-3)², which is not equal to x² - 9. Exam tip: In an area model, the area of a square is the square of its side length.

In the rectangle area model, the side lengths are \(a+4\) and \(a-4\). The partial areas are \(a^2\), \(-4a\), \(4a\), and \(-16\). The terms \(-4a\) and \(4a\) cancel, so the total area is \(a^2-16\). The option \(a^2+16\) misses the negative constant term. Exam tip: use the identity \((x+y)(x-y)=x^2-y^2\).

The total area of the two rectangles is (2\cdot3\cdot2=12). Exam tip: treat the middle part as two equal rectangles.

The square model of ((5+2)^2) gives four parts (5^2), (5\cdot2), (5\cdot2), and (2^2). Exam tip: add all four parts.

In

(a-b)^2=a^2-2ab+b^2

, two rectangular strips of area

ab

are removed from the large square. Their overlapping

b^2

square is removed twice, so it is added back once. Exam tip: the term

-2ab

indicates two strips.

The two equal rectangles have total area (8x), and the small square has area (16). Exam tip: the middle term comes from two rectangles.

The model contains a square with area \\(x^2\\), so one part of its side is \\(x\\). It also contains a small square with area \\(36\\), whose side is \\(6\\), because \\(6^2=36\\). The two rectangles of area \\(6x\\) each are consistent with side lengths \\(x\\) and \\(6\\). Therefore, the outer side is obtained by adding these two side parts: \\(x+6\\).

Option A is correct. The complete model represents \\(x^2+2(6x)+36=x^2+12x+36\\), which is exactly \\((x+6)^2\\). Option B incorrectly doubles the added length, even though there is only one side segment of length \\(6\\). Option C would describe subtraction, and option D mixes a product with a length. Hence the visual pieces and the identity both lead to A.

Each rectangle has area (5a), so the total is (10a). Exam tip: double the equal rectangles.

The four areas in the rectangle model are \(x^2\), \(5x\), \(2x\), and \(10\). Their sum is \(x^2+5x+2x+10=x^2+7x+10\). In option B, the coefficient of \(x\) and the constant term are incorrectly interchanged. Exam tip: combine the like terms carefully: \(5x+2x=7x\).

Two (3y) strips are subtracted, and (9) is added back. Exam tip: keep the middle term negative in ((a-b)^2).

The small square area is subtracted from the large square, and the remainder can be rearranged into a rectangle. Exam tip: link difference of squares with rearrangement.

When the large square is partitioned, it contains two rectangles of dimensions \(a\) by \(b\). Each has area \(ab\), so together they give \(ab+ab=2ab\). The terms \(a^2\) and \(b^2\) represent square regions. Exam tip: identify the mixed term through rectangles.

This is the area of ((x+1)^2). Exam tip: identify (1) as (1^2) and (2x) as two (x) rectangles.

The four parts are (a^2), (7a), (3a), and (21). Exam tip: check the sum and product of the outside numbers.

The side of the square is \(10+2=12\), so its area is \((10+2)^2\). Using the visual identity, \(10^2+2\times10\times2+2^2=100+40+4=144\). Option 124 does not correctly include the combined area \(40\) of the two rectangular parts. Exam tip: in \((a+b)^2=a^2+2ab+b^2\), do not forget the middle term \(2ab\).