Visual models of identities

The total area of the two rectangles is ((x+y)^2-(x^2+y^2)=2xy). Exam tip: subtract the sum of squares from the whole area.

The constant part of the side is (5), so the small corner square is (5^2=25). Exam tip: the last term is the square of the constant part.

In a three-part square, (ab), (bc), and (ca) rectangles appear twice each. Exam tip: count pairwise rectangles.

The constant corner is (1\cdot 7=7) and the (x)-terms are (14x+2x=16x). Exam tip: add the four parts separately.

This is the expansion of ((5x-3y)^2) because the middle term is (-30xy). Exam tip: identify subtraction in the side from the negative middle term.

The total area is ((a+b)^2=144), so the outer side is (12). Exam tip: take square root of area to get the side.

The difference of squares is ((a+b)(a-b)=13\cdot 5=65). Exam tip: no need to find (a) and (b) separately.

The two rectangular regions together have area \(ab+ab=2ab\). Hence the whole square has area \(a^2+2ab+b^2\), equal to \((a+b)^2\). Option B has a negative middle term. Exam tip: two \(ab\) rectangles indicate \(2ab\).

Two (4x) strips are subtracted and the corner (16) is added back. Exam tip: avoid subtracting the overlap twice.

The rectangle’s area is \((x+y)(x-y)\). On multiplying, the middle terms cancel: \(x^2-xy+xy-y^2=x^2-y^2\). Option A would require both sides to be \(x-y\). Exam tip: identify conjugate binomials first.

The missed part is (2ab), made of two (ab) rectangles. Exam tip: never omit the middle rectangles.

In product of sum and difference, the middle terms cancel. Exam tip: remember ((A+B)(A-B)=A^2-B^2).

Each rectangle is (ax=11x), so (a=11) and the small square is (121). Exam tip: identify the constant side from one rectangle.

Two rectangles from (2y) and (3z) appear, so (2\cdot 2y\cdot 3z=12yz). Exam tip: count pairwise rectangles in a three-part square.

(49=7^2) and (2\cdot x\cdot 7=14x), so (k=7). Exam tip: check both square root of last term and middle term.

The area of the whole square is \((p+q)^2\). Adding its parts gives \(p^2+pq+pq+q^2=p^2+2pq+q^2\), so A is correct. Option D misses one \(pq\) rectangle. Exam tip: two equal \(pq\) regions always give the middle term \(2pq\).

The correct model has two (5x) rectangles along with (x^2) and (25). Exam tip: omitting the middle term is a common mistake.

The whole square has side \(p+q\), so its area is \((p+q)^2\). Adding the regions gives \(p^2+pq+pq+q^2=p^2+2pq+q^2\). In \((p-q)^2\), the middle term is negative. Exam tip: combine the two equal \(pq\) regions carefully.

The area of the larger square is \((a+b)^2\), while that of the smaller square is \((a-b)^2\). Therefore, \((a+b)^2-(a-b)^2=(a^2+2ab+b^2)-(a^2-2ab+b^2)=4ab\). In the visual model, the remaining region consists of four parts, each with area \(ab\). \(2ab\) is only the middle term in the expansion of one square, so it is not correct here. Exam tip: expand both squares and cancel the common terms.

The two square regions contribute \(a^2\) and \(b^2\), while the two rectangles together contribute \(2ab\). Hence the total area is \(a^2+2ab+b^2=(a+b)^2\). In \((a-b)^2\), the middle term is negative. Exam tip: count the \(ab\) rectangles to identify the middle term.