Visual models of identities

The side of the square is (a+b), so area is ((a+b)^2). Exam tip: identify the side of the square first.

In ((x+3)^2), the middle term is (2\cdot x\cdot 3). Exam tip: do not forget the two equal rectangles.

In a removal model, the remaining side is (a-b). Exam tip: connect the subtraction square with ((a-b)^2).

The area is ((a+b)(a-b)), which becomes (a^2-b^2). Exam tip: remember product of sum and difference.

The total area is (x^2+4x+4), which is ((x+2)^2). Exam tip: identify the number from unit tiles.

When the square is divided according to lengths \(p\) and \(q\), it forms two rectangles. Each rectangle has area \(p\times q=pq\). Therefore, their combined area is \(pq+pq=2pq\). \(pq\) is the area of only one rectangle, whereas \((p+q)^2\) is the area of the whole square. Exam tip: While adding areas of equal parts in a diagram, always count how many such parts there are.

Removing the two strips gives \(a^2-ab-ab\). The corner \(b^2\) region was removed twice, so it must be added once: \(a^2-2ab+b^2\). Exam tip: add back any overlapping area that was subtracted twice.

The remaining area is (a^2-b^2), which equals ((a+b)(a-b)). Exam tip: understand the cut-and-rearrange model.

The area of the square is \((y+5)^2\). In the area model, there are two rectangles, each with area \(y\times 5=5y\). Their total area, which is the middle term, is \(5y+5y=10y\). \(25\) is the constant term, while \(y^2\) is the first term. Exam tip: in \((a+b)^2=a^2+2ab+b^2\), the middle term is always \(2ab\).

Taking (16) as (4^2), the middle term is (2\cdot x\cdot 4=8x). Exam tip: check the square root of the constant term.

The area of a rectangle is the product of its length and breadth. Here those dimensions are \\(r+2\\) and \\(r+3\\), so the area is \\( (r+2)(r+3)\\). To expand it, multiply each term in the first bracket by each term in the second: \\(r\cdot r=r^2\\), \\(r\cdot3=3r\\), \\(2\cdot r=2r\\), and \\(2\cdot3=6\\). Combining the like middle terms gives \\(r^2+3r+2r+6=r^2+5r+6\\). Therefore, option C is correct. The four terms correspond to the four regions in the rectangle model.

The constant part is produced by multiplying the two constant terms, 2 and 3, giving 6. The coefficient of \\(r\\) comes from the two cross-products, \\(3r\\) and \\(2r\\), which add to \\(5r\\). Option A keeps only the square and constant terms and omits the middle contribution. Option B shows the cross-products but does not combine them or include the constant term, while option D has the wrong coefficients and arrangement. Hence the complete area is \\(r^2+5r+6\\), as stated in option C.

Since ((a-b)^2=a^2-2ab+b^2), the missing part relates to (2ab-b^2). Exam tip: read the removed part carefully.

The area of a square is obtained by multiplying its side by itself. Here the side is \\(2x+1\\), so the area is \\((2x+1)^2\\). Using the identity \\((p+q)^2=p^2+2pq+q^2\\), take \\(p=2x\\) and \\(q=1\\). Then \\(p^2=4x^2\\), \\(2pq=2(2x)(1)=4x\\), and \\(q^2=1\\).

Adding these parts gives \\(4x^2+4x+1\\), so option A is correct. The middle term cannot be omitted: it represents the two equal rectangular parts in the visual model. Option C leaves out that term, while options B and D calculate the first or middle contribution incorrectly. The expansion is therefore confirmed both algebraically and visually.

In the difference of squares model, the rectangle sides are sum and difference. Exam tip: connect (a^2-b^2) with ((a+b)(a-b)).

The area of a square is the square of its side. Thus, \((k-4)^2=k^2-2\cdot k\cdot4+4^2=k^2-8k+16\). In the area model, two rectangular parts of area \(4k\) are removed from the \(k^2\) region, and the corner square of area \(16\) is added. Option A has only \(-4k\) as the middle term, but it should be \(-8k\). Exam tip: in \((a-b)^2=a^2-2ab+b^2\), the middle term is negative.

A square of side (a+b) is divided into these three parts. Exam tip: write the identity by adding areas.

In ((x+1)^2=x^2+2x+1), there are two (x)-rectangle tiles. Exam tip: count rectangles in both directions.

The four products in the rectangle model are \(x^2\), \(x\), \(4x\), and \(4\). The terms containing \(x\) are \(x\) and \(4x\), so their sum is \(x+4x=5x\). Do not combine \(x^2\) with these terms because it has a different exponent. Exam tip: combine only like terms with the same variable and exponent.

The large square has side \(a+b\), so its area is \((a+b)^2\). Adding the four regions gives \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Exam tip: identify both \(ab\) rectangles before choosing the identity.

The large square has area \(a^2\), while the removed square has area \(b^2\); hence the remaining area is \(a^2-b^2\). \((a-b)^2\) represents the area of a smaller square. Exam tip: subtract the area of every removed part.