Visual models of identities

The direct answer is C: y²−6y+9. Apply the identity (a−b)²=a²−2ab+b² with a=y and b=3. Thus (y−3)²=y²−2(y)(3)+3²=y²−6y+9. In a visual or rearranged model, start with the y by y square, remove two strips of area 3y each, and account for the 3 by 3 corner square, which contributes 9. Option A has a positive middle term and therefore matches (y+3)², not (y−3)². Option B has the correct sign but forgets that two cross-regions together make 6y. Option C has all terms correct. Option D incorrectly uses 9y as the middle term and 6 as the constant. Memory cue: for a difference square, the first and last terms are positive, but the middle term is negative: a²−2ab+b².

The direct answer is B: m²−n²=(m+n)(m−n). The large square has area m², and removing the smaller n by n square removes area n². Therefore the remaining area is m²−n². The difference-of-squares identity says this expression can be factored as (m+n)(m−n), which can also be understood by rearranging the remaining pieces into a rectangle with those two side lengths. Option A adds the small square instead of removing it. Option B correctly states both the remaining area and its factorised form. Option C is the identity for (m+n)², where the small square and two mixed rectangles are added. Option D is (m−n)², which represents a complete square with side m−n, not simply a large square with an n² corner removed. Memory cue: removing one square from another gives a difference of squares, not the square of a difference.

The direct answer is A: two rectangles with sides a and b. Partition the large square of side a+b into lengths a and b in both directions. The four regions are an a by a square with area a², two a by b rectangles, each with area ab, and a b by b square with area b². Adding the two equal rectangles gives ab+ab=2ab. Option A is therefore the mixed or cross-term region. Option B represents only the a² part, not 2ab. Option C represents only b². Option D represents the entire large square and has area (a+b)², so it is not one component of the partition. A common confusion is to count only one rectangle and write ab; the coefficient 2 shows that there are two identical rectangles.

The direct answer is A: x+1. A square’s area is the square of its side, so we need to recognise x²+2x+1 as a perfect square. Using (a+b)²=a²+2ab+b² with a=x and b=1 gives (x+1)²=x²+2x+1. Thus the side is x+1. In the visual model, the parts are an x by x square, two x by 1 rectangles, and a 1 by 1 square. Option B gives (x+2)²=x²+4x+4, so its middle and constant terms are wrong. Option C gives (x−1)²=x²−2x+1, with the wrong sign for the middle term. Option D gives (2x+1)²=4x²+4x+1, so even the leading term is wrong. Memory cue: x²+2x+1 is the standard pattern x²+2(x)(1)+1².

The direct answer is C: a²+10a+21. Divide the rectangle into four parts. Their areas are a×a=a², a×7=7a, 3×a=3a, and 3×7=21. Adding them gives a²+7a+3a+21=a²+10a+21. Option A has the correct first two terms but uses 10 instead of the constant product 3×7=21. Option B confuses the product 21 with the coefficient of a and places 10 as the constant. Option C correctly includes every region of the area model. Option D has the correct constant 21 but incorrectly makes the middle term 4a; the cross-terms must be 7a+3a=10a. A useful rule is (a+m)(a+n)=a²+(m+n)a+mn: add the two numbers for the middle coefficient and multiply them for the constant term.

The direct answer is B: 144. First find the side: 10+2=12. The area of a square is side², so the area is 12²=144. Using the visual identity gives the same result: (10+2)²=10²+2(10)(2)+2²=100+40+4=144. The 100 comes from the 10 by 10 square, the 40 comes from two 10 by 2 rectangles, and the 4 comes from the 2 by 2 square. Option A, 120, omits or mishandles part of the area. Option B includes all three regions correctly. Option C, 104, adds only the two square regions and leaves out the middle rectangles. Option D, 124, does not correctly calculate the cross-term 40. Memory cue: in (a+b)², always include the middle term 2ab; adding only a² and b² is not enough.

The correct answer is A: a square with side (u − v). Start with the basic area formula for a square: area = side × side = (side)². If the side length is u − v, then its area is (u − v)(u − v) = (u − v)². This is exactly the visual meaning of the expression. Option B describes a square whose area is (u + v)², because both of its sides are u + v; it represents a different identity. Option C describes a rectangle with area (u + v)(u − v), which simplifies to u² − v², not (u − v)². Option D describes a square with area v², because its side is v. Thus only A has the required side length and gives the required squared expression. A useful memory cue is: for a square, the expression being squared is the length of each side. Do not confuse the square of a difference, (u − v)², with the difference of squares, u² − v².

The direct answer is C: the side of the small square is 6. In a complete-square or area model, x² + 12x + 36 can be compared with x² + 2rx + r². Since 2r = 12, r = 6; also, r² = 36 confirms the result. Algebraically, x² + 12x + 36 = x² + 2(6)x + 6² = (x + 6)². In the visual model, x² represents the large square, the two rectangular parts together represent 12x, and the small corner square has area 36. A square with area 36 has side √36 = 6, taking the positive length. Option A, 12, is the coefficient of x in the middle term, not the side. Option B, 36, is the area of the small square, not its side. Option C is correct because 6² = 36 and 2·6x = 12x. Option D, 3, would produce area 9 and middle term 6x, so it cannot represent the given expression. Memory cue: in x² + 2rx + r², the small-square side is r, the square root of the final term.

The correct answer is A: the middle terms are equal in magnitude and have opposite signs. Multiply the two binomials term by term: (r + s)(r − s) = r·r + r·(−s) + s·r + s·(−s) = r² − rs + rs − s². The two middle terms are −rs and +rs. They represent equal areas or products with opposite signs, so their sum is zero: −rs + rs = 0. What remains is r² − s². In a visual model, one middle rectangular region can be viewed as a positive rs area and the other as a negative rs area; they cancel each other. B is incorrect because r² is generally not zero. C is incorrect because s² is the final negative term, not the reason the middle terms cancel. D is incorrect because cancellation depends on opposite signs, not on the two side lengths being equal. The key memory cue is that conjugate factors, (r+s) and (r−s), always create opposite middle products. This gives the identity (r+s)(r−s)=r²−s².

The correct answer is C: x² + 9x. The area of a rectangle is one side multiplied by the other side. Apply the distributive law to x(x + 9): multiply x by each term inside the bracket. Thus x·x + x·9 = x² + 9x. In a visual area model, the side x + 9 is split into two parts, x and 9. The rectangle with dimensions x by x has area x², while the rectangle with dimensions x by 9 has area 9x. Adding the two component areas gives x² + 9x. Option A is wrong because x·9 is 9x, not 9; the variable x cannot simply disappear. Option B reverses the roles and incorrectly creates 9x². Option D gives only the sum of the side lengths, not the area; area requires multiplication. The memory cue is: when a factor outside brackets multiplies a sum, multiply it by every term inside: a(b+c)=ab+ac. The rectangle picture is a geometric way to understand this distributive rule.

The correct answer is A: 4x² + 4x + 1. A square has equal sides, so its area is side × side. Here the side is 2x + 1, and therefore the area is (2x + 1)². Use the identity (a+b)² = a² + 2ab + b², with a = 2x and b = 1. Then a² = (2x)² = 4x², 2ab = 2(2x)(1) = 4x, and b² = 1. Adding these parts gives 4x² + 4x + 1. The visual model shows one square of area 4x², two rectangular pieces whose combined area is 4x, and one small square of area 1. Option B is wrong because it changes (2x)² into 2x². Option C is wrong because it uses only 2x instead of the two equal cross-products, which together make 4x. Option D omits the middle rectangles. The memory cue is “first square, twice the product, last square”: a²+2ab+b².

Direct answer: A, (x + y)². Imagine dividing a large square whose side is x + y into four parts. The x-by-x part has area x², the y-by-y part has area y², and the two remaining rectangles have areas xy and xy. Adding all four areas gives x² + xy + xy + y² = x² + 2xy + y². This is exactly the identity (x + y)². Option A is correct because it includes the two positive middle rectangles. Option B, (x − y)², represents a subtraction model and has negative middle terms, not two positive xy rectangles. Option C, x² − y², represents the difference of two squares and has no 2xy term. Option D, x² + y², includes only the two square areas and leaves out both rectangles. Memory cue: a plus binomial square has the pattern first square + twice the product + second square.

Direct answer: C, 25. In the visual model for (x − 5)², the number 5 represents the length of the small side associated with the constant part. A square’s area is side × side, so this small square has area 5 × 5 = 25. Algebraically, (x − 5)² = x² − 10x + 25; the final constant term comes from squaring 5. Option A, 25x², incorrectly contains x² and would represent neither the constant square nor its area. Option B, 10x, is the combined middle term in magnitude, produced by two x-by-5 rectangles; it is not the small square’s area. Option C is correct because 5² = 25. Option D, x², is the area of the large x-by-x square, not the small constant square. Memory cue: the minus sign affects the middle term, but the square of 5 is positive.

Direct answer: A, a square with side (a + b). Divide each side of this square into lengths a and b. The resulting four regions are an a-by-a square with area a², two rectangles of dimensions a and b with areas ab and ab, and a b-by-b square with area b². Their total area is a² + ab + ab + b² = a² + 2ab + b². Therefore option A displays all four parts directly. Option B, the square with side (a − b), represents a² − 2ab + b², so its middle regions are associated with subtraction. Option C has area (a+b)(a−b) = a² − b² and therefore does not contain the 2ab term. Option D has area (ab)² and does not split into the required four regions. Memory cue: the plus square is the four-tile model with two positive ab rectangles.

Direct answer: B, (x + 4) and (x + 5). Multiply the proposed dimensions: (x + 4)(x + 5) = x² + 5x + 4x + 20 = x² + 9x + 20. A rectangle model with sides split into x and 4 in one direction and x and 5 in the other produces exactly four regions: x·x = x², x·5 = 5x, 4·x = 4x, and 4·5 = 20. Option A gives (x+9)(x+20), which produces x² + 29x + 180, not the listed parts. Option B matches every part and is correct. Option C gives x² + 12x + 20, so its middle regions do not match 4x and 5x. Option D gives x² + 21x + 20 and also has the wrong middle terms. Memory cue: pair the two x-terms with the two constant side lengths; the constants must multiply to the corner area 20.

Direct answer: A, two rectangles, each of area 3x. A square with side x+3 is divided into an x-by-x square, two x-by-3 rectangles, and a 3-by-3 square. Their areas are x², 3x, 3x, and 9. Adding them gives x² + 3x + 3x + 9 = x² + 6x + 9. The student kept x² and 9 but omitted both middle rectangles, whose combined area is 6x. Option A is therefore correct. Option B is wrong because there is no single 9x rectangle; the two middle regions are each 3x. Option C is wrong because the model contains only one x² square. Option D is wrong because a line segment is a length, not one of the area regions, and it does not account for the missing 6x term. Memory cue: in (x+a)², never forget the two cross-products, giving 2ax.

The direct answer is C: a² + 2ab + b² = (a+b)². Imagine a large square whose side is a+b. Divide each side into lengths a and b. The large square then contains one square of area a², one square of area b², and two rectangles, each having area ab. Adding the four inner areas gives a²+ab+ab+b² = a²+2ab+b². The area calculated from the outside is (a+b)², so both expressions are equal. A is the difference-of-squares identity and needs a square with a difference of lengths. B describes a single rectangle formed by lengths a and a+b, not the complete four-part square. C exactly matches the visual partition. D is not generally true because (a−b)² equals a²−2ab+b², not a²−b². Memory cue: a square of a sum has two identical middle rectangles, hence 2ab.

The direct answer is D: 49. The area of any square is side × side, so here it is (8−1)². Use the identity (a−b)² = a²−2ab+b² with a=8 and b=1: (8−1)² = 8²−2(8)(1)+1² = 64−16+1 = 49. We can also check directly: 8−1=7 and 7²=49. A, 48, is not the square of the side and does not result from the identity. B, 63, may arise from an incomplete subtraction but is not the area. C, 64, is only 8²; it ignores that 1 was subtracted from the side. D is correct because it includes the square of the first term, the negative middle term, and the square of the second term. Memory cue: in (a−b)², the middle term is −2ab and the last term remains positive.

The direct answer is A. A rectangle with length a+b and breadth a has area a(a+b). If the length is split into parts a and b, the rectangle is divided into an a by a square and an a by b rectangle. Their areas are a² and ab, so the total is a²+ab. Factoring out a gives a(a+b), exactly as required. B represents a much larger square with area (a+b)² and would contain four parts, not just a² and ab. C has area b(a−b), which is a different expression and uses the wrong common dimension. D has side ab, so its area is (ab)², not a(a+b). Thus A is the only model whose dimensions and partition match. Memory cue: a common factor a means one side of the rectangle is a; the other side is the sum a+b.

The direct answer is C: x²−16. The rectangle has side lengths x+4 and x−4, so its area is their product. Apply the identity (a+b)(a−b)=a²−b² with a=x and b=4: (x+4)(x−4)=x²−4²=x²−16. Expanding confirms this: x·x + x(−4) + 4x + 4(−4) = x²−4x+4x−16 = x²−16. The middle terms cancel because they are opposites. A, x²+16, incorrectly adds the two squares instead of subtracting them. B is the expansion of (x−4)², not a product of conjugate factors. D is the expansion of (x+4)², not the given product. Therefore the rectangle area is C. Memory cue: same first term with opposite second terms means difference of squares.

The direct answer is B: 4. In the visual model for (3x+2)², the large square is divided using lengths 3x and 2. The small corner square has side 2, so its area is 2×2=4. A, 2, is only the side length, not the area. B is correct because the area of a square is the square of its side. C, 6x, is the area of each mixed rectangle with sides 3x and 2, since (3x)(2)=6x; it is not the small corner square. D, 9x², is the area of the large 3x by 3x square. The complete identity is (3x+2)² = 9x²+12x+4, where the two mixed rectangles together contribute 12x. Thus the constant corner part must be 4. Memory cue: in a two-part square model, square each piece for the corner squares and multiply unlike pieces for the rectangles.

The direct answer is C: x−5. Recognize the expression as a perfect square. The first term x² is x², the last term 25 is 5², and the middle term is −10x = −2·x·5. Therefore x²−10x+25 = x²−2·x·5+5² = (x−5)². In a visual model, this means the entire square has side x−5. A, x+5, would produce (x+5)²=x²+10x+25, with a positive middle term. B, x−10, would produce x²−20x+100, so both its middle and constant terms are wrong. D, x+25, would produce x²+50x+625, which is also unrelated. The negative middle term is the key clue: the side must contain subtraction. Memory cue: take the square root of the last term and use the sign of the middle term.

The direct answer is A: dimensions 6+2 and 6−2, which are 8 and 4. The difference-of-squares identity says a²−b²=(a+b)(a−b). Substituting a=6 and b=2 gives 6²−2²=(6+2)(6−2)=8×4=32. Direct calculation also gives 36−4=32, confirming the same area. B gives dimensions 6 and 2, whose area is only 12, so it does not represent the difference of squares. C gives 12 and 4, whose product is 48 and incorrectly doubles both original numbers. D gives 4 and 4, whose area is 16 and omits the required sum factor. Thus A is the only pair that preserves the original area. Memory cue: subtracting two squares becomes a rectangle whose sides are the sum and the difference of the square roots.

The direct answer is A. Begin with a square whose side is a+b. Divide each side into segments a and b. This creates four regions: an a by a square with area a², a b by b square with area b², and two rectangles, each with dimensions a by b and area ab. Adding them gives a²+ab+ab+b²=a²+2ab+b², which equals (a+b)². B describes only one rectangle and gives ab, so it misses most of the large square. C concerns a difference, a−b, and does not produce the sum-square identity; its parts would not all be equal in general. D uses a triangle, whose area would be one-half base times height, not the area of the square. Therefore A is correct. Memory cue: the square of a two-term sum always has two equal cross-rectangles.

The direct answer is B: twice. Split each side of the large square into lengths a, b, and c. For every pair of different lengths, two rectangular regions occur because the pair can be arranged in two orientations. For the pair b and c, one rectangle has dimensions b by c in one position and another has dimensions c by b in the symmetric position. Both have area bc, so the total contribution is 2bc. A is wrong because it counts only one orientation. C is wrong because the pair b,c does not create three regions. D counts all ordered pairs of three letters or all six possible arrangements, but the visual square has only one pair of bc rectangles. The full expansion is (a+b+c)²=a²+b²+c²+2ab+2bc+2ca. Memory cue: each two-letter cross-term in a square expansion appears twice.