Visual models of identities

In the area model, the side parts are \(3x\) and \(2\). The two rectangular regions have areas \(3x\times2=6x\) and \(2\times3x=6x\). Therefore, the total area of the x-term parts is \(6x+6x=12x\). \(9x^2\) is the area of the \(3x\times3x\) square, so it is not an x-term. Exam tip: in a square area model, remember to add both equal middle rectangles.

(25x^2=(5x)^2), (9=3^2), and the middle term is (-2\cdot5x\cdot3=-30x). Exam tip: take roots of square terms and match the sign.

The cross regions are (ab,ba,ac,ca,bc,cb), making (6) rectangles. Exam tip: count the off-diagonal sections.

Three squares and six cross rectangles give (a^2+b^2+c^2+2ab+2bc+2ca). Exam tip: multiply equal pairs by (2).

After rearrangement, the rectangle has length \(x+y\) and breadth \(x-y\). Its area is \((x+y)(x-y)=x^2-y^2\). Exam tip: match the area left after removing the small square with the product of the rectangle’s sides.

((-5)+(-8)=-13) and ((-5)(-8)=40). Exam tip: with negative rectangles, the constant term can be positive.

The square’s area is the sum of its four parts: \(p^2+pq+pq+q^2=p^2+2pq+q^2\), so A is correct. A \((p-q)^2\) model involves subtraction. Exam tip: combine the two \(pq\) rectangles to get \(2pq\).

\(12^2-5^2=(12-5)(12+5)=7\cdot17\). Exam tip: form sum and difference with the removed square.

The large square has side \(a+b\), so its area is \((a+b)^2\). Splitting each side into lengths \(a\) and \(b\) forms one \(a^2\) square, one \(b^2\) square, and two rectangles of area \(ab\) each. Thus, the total area is \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Option C has only one \(ab\) rectangle, so it represents \(a^2+ab+b^2\). Exam tip: count the two equal \(ab\) rectangles separately in an area model.

(16a^2=(4a)^2), (25b^2=(5b)^2), and (2\cdot4a\cdot5b=40ab). Exam tip: make sides from square terms.

The constant square has side (4), so its area is (4^2=16). Exam tip: observe the small corner square separately.

A square of side \\(x+y\\) has total area \\((x+y)^2\\). Splitting it into its visual parts gives one \\(x^2\\) square, one \\(y^2\\) square, and two rectangles, each with area \\(xy\\). Thus the whole area is \\(x^2+xy+xy+y^2=x^2+2xy+y^2\\).

The question says that the two square parts together have area 50, so \\(x^2+y^2=50\\). It also says that the two rectangles together have area 48, so \\(2xy=48\\). Adding all four visible parts gives \\(50+48=98\\). Therefore option C is correct. Option B gives only the rectangular parts and does not include the two square parts; option D multiplies quantities that should be added.

The small square has side (6), so the added-back area is (6^2=36). Exam tip: remember to add once the corner removed twice.

First (x^2+2xy+y^2=(x+y)^2), then (z^2) is subtracted. Exam tip: identify the perfect square first.

Dividing the square in both directions at \(p\) and \(q\) gives \(p\times p=p^2\), \(q\times q=q^2\), and two \(p\times q\) regions. Thus the area is \(p^2+2pq+q^2\). Exam tip: count both \(pq\) rectangles.

The two strips have total area \(ab+ab=2ab\), but their \(b^2\) corner is subtracted twice. Hence the remaining area is \(a^2-2ab+b^2=(a-b)^2\). Exam tip: add back the overlapping region once.

((2x+3)(x+1)=2x^2+5x+3) is not correct, but ((2x+1)(x+3)=2x^2+7x+3) is correct. Exam tip: expand all options to check.

The initial area is \(a^2\). Removing two strips subtracts \(2ab\), but their overlap is removed twice, so add \(b^2\): \(a^2-2ab+b^2=(a-b)^2\). Exam tip: always check an overlapping region separately.

Dividing a square of side \((a+b)\) gives one \(a^2\) square and one \(b^2\) square. The remaining two rectangles each have area \(ab\), producing \(2ab\). Exam tip: always look for two \(ab\) rectangles.

The total area of the square is \((a+b)^2\). Adding its four parts gives \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Option B has a negative middle term. In exams, remember to count both \(ab\) rectangles.