Visual models of identities

When the side of the square is split into \(4x\) and \(3\), two mixed rectangles are formed. Each rectangle has area \(4x\times 3=12x\). Therefore, their total area is \(12x+12x=24x\). \(12x\) is the area of only one mixed strip. Exam tip: in the visual model of \((a+b)^2\), the combined mixed area is always \(2ab\).

The area is split into four parts: \(2x\cdot x=2x^2\), \(2x\cdot5=10x\), \(1\cdot x=x\), and \(1\cdot5=5\). Thus, the total area is \(2x^2+10x+x+5=2x^2+11x+5\). Option A misses the \(x\) term. Exam tip: in an area model, multiply every term of one side by every term of the other side, then combine like terms.

In the rectangle model, the constant corner is obtained by multiplying the constant terms from the two sides. Here, the constant terms are 2 and 4, so the constant corner is \(2\times4=8\). The value 12 comes from multiplying \(3y\) and 4, so it is not a constant term. Exam tip: to find the constant term, multiply only terms that contain no variable.

The linear parts are (12y) and (2y), so the total is (14y). The correct expansion is \(3y^2+14y+8\).

The equal but opposite (10x) parts cancel. In exams, the linear terms cancel in conjugate sides.

The area of a square is the square of its side. Hence, the area of a square with side \((5a+2)\) is \((5a+2)^2\). Using \((x+y)^2=x^2+2xy+y^2\), we get \((5a+2)^2=25a^2+2(5a)(2)+4=25a^2+20a+4\). Therefore, option B is correct. Option A misses the required factor 2 in the middle term. Exam tip: when squaring a binomial, include both squared terms and the middle term \(2xy\).

The area of the square is the square of its side: \((2p-7)^2\). Using \((a-b)^2=a^2-2ab+b^2\), with \(a=2p\) and \(b=7\), gives \(4p^2-28p+49\). In the visual model, two \(2p\times 7\) strips are removed, so the middle term is \(-28p\). Option A has only half of the required middle term. Exam tip: the middle term in \((a-b)^2\) is always negative.

Dividing the large square into lengths \(a\) and \(b\) gives squares of areas \(a^2\) and \(b^2\). The remaining two rectangles have areas \(ab+ab=2ab\). Exam tip: identify the mixed term as two equal rectangles.

The total area is \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Since the side of the complete square is \(a+b\), its area is \((a+b)^2\). Exam tip: count both \(ab\) rectangles; this creates the middle term \(2ab\).

Both strips together give (6k), and (6k=24). In exams, connect combined strips with the middle term.

The border area is ((x+5)^2-x^2=10x+25). In exams, subtract the inner area from the outer area.

Both sides are equal, so it is a square of side (x+6). In exams, treat equal sides as a square model.

One mixed rectangle is (2ab), and there are two such rectangles. In exams, count the mixed rectangle twice.

In the standard visual model, the small square has side \(2b\). Hence, its area is \((2b)^2=4b^2\). The expression \(2ab\) is the area of a rectangle with sides \(a\) and \(2b\), while \(a^2\) represents the area of a square with side \(a\). Exam tip: find the area of a square by squaring its side length.

The area of a square is the square of its side. For side \\(2a-b\\), the area is \\((2a-b)^2\\). Using the identity \\((x-y)^2=x^2-2xy+y^2\\), take \\(x=2a\\) and \\(y=b\\). Then \\(x^2=4a^2\\), \\(2xy=2(2a)(b)=4ab\\), and \\(y^2=b^2\\). Hence the expansion is \\(4a^2-4ab+b^2\\).

The middle term is negative because the binomial contains subtraction. Its coefficient is 4, not 2, because the two equal cross-rectangles together contribute \\(2(2a)(b)=4ab\\). Therefore, option A is correct. Option D would apply to \\((2a+b)^2\\), while the other expressions do not follow the square identity.

The area of the rectangle is the product of its side lengths: \((3x+2)(3x-2)\). It has the form \((a+b)(a-b)=a^2-b^2\), where \(a=3x\) and \(b=2\). Hence, the area is \((3x)^2-2^2=9x^2-4\). \(9x^2+4\) incorrectly uses the sum of squares; this identity requires their difference. Exam tip: when matching binomials have opposite signs, apply the difference-of-squares identity directly.

A perfect-square trinomial has the form \(x^2+2ax+a^2=(x+a)^2\). In option A, \(10x=2\times5\times x\) and \(25=5^2\), so \(x^2+10x+25=(x+5)^2\). Its tiles can form a square using one \(x^2\) tile, 10 \(x\)-tiles, and 25 unit tiles. In option B, the constant term would need to be 25, not 20. Exam tip: halve the middle coefficient and square the result.

The areas of the squares are \((x+1)^2\) and \((x-1)^2\). Thus, their difference is \((x+1)^2-(x-1)^2\). Applying the identity for the difference of squares gives \([(x+1)-(x-1)]\,[(x+1)+(x-1)]=2\times 2x=4x\). The expression \(x^2-1\) is the product \((x+1)(x-1)\), not the difference between the areas. Exam tip: For an area difference, write the larger square's area first and then use \(a^2-b^2\).

The remaining area is ((x+4)^2-x^2=8x+16). In exams, find border area using area difference.

A square whose side is split into parts of lengths \(a\), \(b\), and \(c\) contains three small square regions: one with side \(a\), one with side \(b\), and one with side \(c\). The area of a square is side multiplied by itself, so these three areas are \(a^2\), \(b^2\), and \(c^2\). The question asks only for the small squares, not for the rectangular regions between them.

Adding the three square areas gives \(a^2+b^2+c^2\). The terms \(ab\), \(bc\), and \(ca\) describe rectangular areas, and their doubled sum belongs to the cross terms in the full expansion of \((a+b+c)^2\). Therefore, option A correctly represents the total area of only the small squares. The complete model would also include the rectangular parts, but they must not be included here.