Visual models of identities

Correct answer: B. 20t

Correct answer: B, 20t. Think of the side 5t-2 as a long length 5t with a part of length 2 removed. In the square model, the removed part creates two identical rectangular strips. Each strip has dimensions 5t and 2, so each has area (5t)(2)=10t. Because there are two such strips, their combined area is 10t+10t=20t. This agrees with the identity (5t-2)^2=(5t)^2-2(5t)(2)+2^2=25t^2-20t+4. Option A is only the area of one strip. Option C is the large positive square (5t)^2, not the subtracted strips. Option D is the small corner square 2^2. Memory cue: in (p-q)^2, the two side strips together give 2pq and are subtracted.

Correct answer: D. 2a^2-a-15

Correct answer: D, 2a^2-a-15. Split the first side into 2a and 5 and the second side into a and -3. The four signed regions are (2a)(a)=2a^2, (2a)(-3)=-6a, 5(a)=5a, and 5(-3)=-15. Add them carefully: 2a^2-6a+5a-15=2a^2-a-15. The negative sign comes from the -3 part, so the second rectangle is a negative contribution. Option A incorrectly combines the linear terms as +2a. Option B adds -6a and 5a with the wrong sign and gets +11a. Option C omits the -15 corner and uses only one linear contribution. Option D includes all four parts with their correct signs. A safe check is ordinary multiplication: (2a+5)(a-3)=2a^2-6a+5a-15.

Correct answer: D. 7y

The direct answer is D: each removed strip has area 7y. Start with the perfect-square identity (y-7)^2=y^2-14y+49. In a square model, the large square has area y^2, the small corner square has area 7^2=49, and two equal rectangular strips are removed. Each strip has one side y and the other side 7, so its area is y multiplied by 7, which is 7y. The two strips together have area 7y+7y=14y, producing the middle term -14y. A, y^2, is the area of the original large square, not one strip. B, 14y, is the combined area of both strips. C, 49y incorrectly multiplies the constant square area by y. D is correct. Memory cue: in (y-k)^2, the middle term is -2ky, so each of the two equal strips has area ky.

Correct answer: A. 12mn

The direct answer is A: the combined area is 12mn. Divide each side of the square into parts 2m and 3n. The square model contains a (2m) by (2m) square, a (3n) by (3n) square, and two mixed rectangles. Each mixed rectangle has dimensions 2m and 3n, so its area is (2m)(3n)=6mn. Since there are two identical mixed rectangles, their combined area is 6mn+6mn=12mn. This is also the middle term in (2m+3n)^2=(2m)^2+2(2m)(3n)+(3n)^2. B gives only one rectangle. C adds the two square areas and ignores the mixed rectangles. D does not result from multiplying either pair of side lengths. Thus A is correct. Memory cue: the mixed part of (p+q)^2 is 2pq, not pq.

Correct answer: B. 9a^2-12ab+4b^2

The direct answer is B. Use the identity (p-q)^2=p^2-2pq+q^2 with p=3a and q=2b. The first square is (3a)^2=9a^2. The two mixed rectangles together have area 2(3a)(2b)=12ab, and because this is a difference, that middle part is subtracted. The small corner square has area (2b)^2=4b^2 and is added. Therefore (3a-2b)^2=9a^2-12ab+4b^2. A has the wrong middle coefficient because it forgets one part of the factor 2. C squares the terms incorrectly. D has a positive middle term, which belongs to a sum such as (3a+2b)^2, not a difference. A useful warning is that the sign of the middle term follows the sign between the two terms, while its coefficient comes from doubling their product.

Correct answer: C. \(x^2-4y^2\)

Area of the rectangle is the product of its side lengths:
\[ (x+2y)(x-2y). \]
This is a difference of squares: \( (x)^2-(2y)^2 \). So
\[ (x+2y)(x-2y)=x^2-(2y)^2=x^2-4y^2. \]
Hence option C.

Correct answer: D. ab

The direct answer is D: each rectangle has area ab. Apply the difference-of-squares identity with A=a+b and B=a-b: A^2-B^2=(A-B)(A+B). The first factor is (a+b)-(a-b)=2b, and the second is (a+b)+(a-b)=2a. Hence the total difference is (2b)(2a)=4ab. The visual proof says this total area is rearranged into four equal rectangles, so each one has area 4ab divided by 4, namely ab. A is the total area, not the area of one rectangle. B would be half the total and does not fit four equal parts. C is the difference of the separate squares a^2 and b^2, not this expression. D is correct. Memory cue: when a total area is split into four equal pieces, divide by four after simplifying the total.

Correct answer: A. 2xy+2yz+2zx

The direct answer is A. Expand the square of three terms by viewing the model as nine regions: three square regions x^2, y^2, z^2 and six mixed rectangles. The expansion is (x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2zx. If only the sum of the three square regions, x^2+y^2+z^2, is removed, the six mixed rectangles remain. The two rectangles involving x and y contribute 2xy; those involving y and z contribute 2yz; and those involving z and x contribute 2zx. Their total is 2xy+2yz+2zx. B forgets that every pair appears twice. C is the sum of side expressions, not an area. D has three variables multiplied together and is not produced by squaring a sum. Memory cue: in a three-term square, every pair contributes twice.

Correct answer: B. 2ab

The governing concept is the area model for the identity (a + b)² = a² + 2ab + b². The complete square has side a + b, so its area is (a + b)². Within the model, one square contributes a² and another contributes b². The two remaining regions are rectangles, each with side lengths a and b, so each has area ab. Their combined area is therefore ab + ab = 2ab. Equivalently, subtracting the two known square areas from the whole gives (a+b)² − a² − b² = a² + 2ab + b² − a² − b² = 2ab. Hence option B is correct. Option A gives only one rectangle, C belongs to a difference-of-squares expression, and D represents a length rather than an area.

Correct answer: D. 8x

The direct answer is D: the result is 8x. Use A^2-B^2=(A-B)(A+B), taking A=2x+1 and B=2x-1. Then A-B=(2x+1)-(2x-1)=2, because the 2x terms and signs simplify, while A+B=(2x+1)+(2x-1)=4x. Multiplying gives 2×4x=8x. A direct check confirms this: (2x+1)^2=4x^2+4x+1 and (2x-1)^2=4x^2-4x+1; subtraction cancels 4x^2 and 1, leaving 8x. A misses the factor 2. B is only one factor, not the product. C is the result of confusing the difference with one of the expanded square forms. D is correct. Common warning: subtracting a negative term changes it to addition, so 4x-(-4x)=8x.

Correct answer: C. (3x + 1) and (x + 3)

The direct answer is C: the rectangle sides are 3x + 1 and x + 3. In a tile model, the product of the side lengths must reproduce the complete polynomial. Multiply term by term: (3x + 1)(x + 3) = 3x·x + 3x·3 + 1·x + 1·3 = 3x² + 9x + x + 3 = 3x² + 10x + 3. Thus every tile region is accounted for: one x² region, ten x regions in total, and one constant region of 3. A is wrong because (3x + 10)(x + 3) = 3x² + 19x + 30. B is wrong because (3x + 3)(x + 1) = 3x² + 6x + 3. C matches all coefficients. D gives x² + 10x + 9, so even its leading and constant terms differ. Memory cue: check first-term product, cross-term sum, and constant product.

Correct answer: D. 8x + 32

The direct answer is D: the remaining area is 8x + 32. The difference of two square areas is (x + 6)² − (x + 2)². Use the identity p² − q² = (p + q)(p − q), taking p = x + 6 and q = x + 2. Then the difference is [(x + 6) + (x + 2)] [(x + 6) − (x + 2)] = (2x + 8)(4) = 8x + 32. A, 4x + 16, is only half the correct result and does not use the full side difference. B, 4x + 32, has an incorrect x-coefficient. C, 8x + 16, has the wrong constant term. D correctly represents the total border-like area between the two squares. As a check, direct expansion gives x² + 12x + 36 − (x² + 4x + 4) = 8x + 32. Memory cue: difference of squares equals sum times difference.

Correct answer: A. 12x + 24

The direct answer is A: the border area is 12x + 24. The border is the part inside the outer square but outside the inner square, so subtract the two areas: (x + 5)² − (x − 1)². Expand carefully: (x + 5)² = x² + 10x + 25, and (x − 1)² = x² − 2x + 1. Therefore, the border area is x² + 10x + 25 − x² + 2x − 1 = 12x + 24. A is correct. B and C have incorrect coefficients of x; they result from incomplete or mistaken subtraction. D is only the area of the outer square, so it includes the inner square and is not the border alone. The same result follows from the difference-of-squares identity: [(x + 5) + (x − 1)] [(x + 5) − (x − 1)] = (2x + 4)(6) = 12x + 24. Memory cue: border area means outside area minus inside area.

Correct answer: D. 13

The direct answer is D: x = 13. The final square has side x − 4 and area 81, so (x − 4)² = 81. A square side represents a positive length, therefore x − 4 = 9 rather than −9. Adding 4 to both sides gives x = 13. The given added-back corner 16 is consistent with the correction term 4² = 16, but the final-area equation is sufficient to find x. A, x = 5, gives side 1 and area 1. B, x = 9, gives side 5 and area 25. C, x = 17, gives side 13 and area 169. D alone gives side 9 and area 81. Pure algebra would also show the second equation x − 4 = −9, giving x = −5, but that cannot represent a positive square side in this geometric model. The safe procedure is: take the positive square root for a length, then solve the linear equation.

Correct answer: A. 2

The direct answer is A: the difference between the areas is 2. The original rectangle has area (x + 2)(x + 5). Multiplying gives x² + 7x + 10. The new rectangle has area (x + 3)(x + 4), which gives x² + 7x + 12. Subtract the original area from the new area: (x² + 7x + 12) − (x² + 7x + 10) = 2. The x² terms cancel and the 7x terms also cancel, so only the constant difference remains. Option A is correct because the new area is exactly 2 square units greater. Option B is wrong because the constant terms 12 and 10 are not equal. Option C is wrong because the x terms cancel rather than leaving x. Option D is wrong because no 2x term remains after subtraction. A useful check is to notice that the two products have the same sum of factors, 7x, but their constant products are 10 and 12. Memory cue: when comparing expanded expressions, combine like terms and cancel identical terms carefully.

Correct answer: B. ac + ad + bc + bd

The direct answer is B: (a + b)(c + d) = ac + ad + bc + bd. In the area model, one side of the large rectangle is split into lengths a and b, while the other side is split into c and d. This creates four small rectangles. Their areas are a·c = ac, a·d = ad, b·c = bc, and b·d = bd. Adding them gives the total area ac + ad + bc + bd. A is wrong because ab and cd multiply pieces from the same side and do not represent the four rectangles. C incorrectly squares each separate symbol; the expression is a product of two binomials, not a sum of four squares. D is a product of all four symbols and does not represent the sum of the four subareas. Option B is exactly the distributive property applied twice. Memory cue: choose one segment from each side for every small rectangle.

Correct answer: C. (x + 5)² is larger by 9

The direct answer is C: (x + 5)² is larger by 9. Expand the rectangle product: (x + 2)(x + 8) = x² + 8x + 2x + 16 = x² + 10x + 16. Expand the square: (x + 5)² = x² + 10x + 25. The x² terms and the 10x terms are identical, so they cancel when the areas are compared. The remaining difference is 25 − 16 = 9. A reverses which expression is larger. B is wrong because the constant terms differ. D is wrong because the linear terms are equal, so no 3x remains in the difference. Option C is therefore correct. A useful visual interpretation is that both models contain the same large x² region and the same total pair of x-strips; only the constant corner differs. Memory cue: when matching terms cancel, compare the remaining constants.

Correct answer: A. (x-k)

The direct answer is A: each side is x-k. Start with the standard identity (x-k)^2 = x^2 - 2kx + k^2. In a visual square model, the large x by x square contributes x^2. Two strips of dimensions x by k are removed, so together they contribute -2kx. Removing both strips removes the small k by k corner twice, so that corner must be added back once; its area is k^2. Therefore the complete expression is exactly (x-k)^2, and the side of the square is x-k. Option A matches the square, the two negative strips, and the restored corner. Option B gives (x+k)^2 = x^2+2kx+k^2, so its middle sign is wrong. Option C gives 4x^2-4kx+k^2, not the given expression. Option D gives x^2-4kx+4k^2, so both the middle and final terms are wrong. Memory cue: a negative middle term means the quantity inside the square has a minus sign.

Correct answer: B. (2xy)

The direct answer is B: the combined rectangle has area 2xy. Each original rectangle has one side x and the other side y, so its area is x multiplied by y, or xy. There are two such rectangles, and they are joined without overlap or a gap. Area is additive, so their total area is xy+xy=2xy. Changing the arrangement can make a longer rectangle, but it cannot change the amount of surface covered. This is also the middle part of the identity (x+y)^2 = x^2+2xy+y^2. Option A counts only one of the two rectangles. Option B correctly counts both rectangles. Option C is the sum of the two square regions, x^2 and y^2, rather than the mixed rectangles. Option D adds lengths or perimeter-like quantities; it is not an area expression and does not represent two xy regions. A useful check is to count pieces first: two pieces, each of area xy, give 2xy.

Correct answer: C. (2ca)

The direct answer is C: the missing part is 2ca. To expand a square of three terms, multiply every term by every term. The three self-products are a^2, b^2, and c^2. Each pair of different terms appears twice: ab and ba give 2ab; bc and cb give 2bc; ca and ac give 2ca. Thus (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca. The model already contains the three square regions and the 2ab and 2bc rectangles, so the remaining pair of mixed rectangles must have total area 2ca. Option A is incomplete because it includes each mixed product only once. Option B is a degree-three product and cannot arise from squaring a sum. Option C is exactly the two a-by-c rectangles. Option D is only a linear expression, not a collection of quadratic area parts. Memory cue: for three terms squared, write three squares plus twice each pair: ab, bc, and ca.

Correct answer: D. (x^2+8x)

The direct answer is D: the remaining area is x^2+8x. The full square has side x+4, so its area is (x+4)^2 = x^2+8x+16. In the visual model, the x-by-x central square contributes x^2. The two strips each have area 4x, so together they contribute 8x. The small corner has side 4 and area 4×4=16. Since that 16 corner is removed while both strips are retained, subtract only 16 from the total: x^2+8x+16-16=x^2+8x. Therefore D is correct. B keeps only one 4x strip and misses the second strip. C keeps the corner but leaves out the two strips. A is not the correct result of the square expansion and corner removal. The visual check is simple: after removing the corner, the remaining pieces are one x^2 square and two 4x strips.

Correct answer: A. (28x)

The direct answer is A: the two missing strips together have area 28x. Use the square identity (p+q)^2=p^2+2pq+q^2 with p=2x and q=7. The large square contributes (2x)^2=4x^2, and the small corner contributes 7^2=49. Each mixed strip has dimensions 2x by 7, so each has area 14x. There are two such strips, one from each direction, and their total area is 14x+14x=28x. Equivalently, the middle term is 2(2x)(7)=28x. Option A counts both strips. Option B counts only one strip, so it is half the needed middle area. Option C uses 7x and misses the factor 2x in the product. Option D does not come from the identity and cannot represent the two equal mixed rectangles. A useful memory cue is: the middle area is twice the product of the two side labels, not just one product.

Correct answer: A. (4kx)

The direct answer is A: the difference is 4kx. Let the larger square area be (x+k)^2 and the smaller one be (x-k)^2. Expand them separately: (x+k)^2=x^2+2kx+k^2 and (x-k)^2=x^2-2kx+k^2. Subtract the second from the first. The equal x^2 terms cancel, and the equal k^2 terms also cancel. The remaining difference is 2kx-(-2kx)=4kx. In a visual model, the common x^2 region and common k^2 corner disappear in the comparison; the cross-strips produce the difference. Option A is correct. Option B notices only one cross contribution and misses the opposite signed term from the second square. Option C is the difference of the side lengths' product pattern, not the difference of these two areas. Option D adds the common square contributions instead of subtracting them. The identity (x+k)^2-(x-k)^2=4kx is a reliable memory check.

Correct answer: A. 2x + 3y

The correct answer is A: 2x + 3y. In a square area model, the two pure-square parts tell us the two pieces of the side. Since 4x² = (2x)², one side piece is 2x. Since 9y² = (3y)², the other side piece is 3y. Therefore the complete outer side is 2x + 3y. We can check this by squaring: (2x + 3y)² = 4x² + 12xy + 9y². The two middle rectangles each have area 6xy, so together they give 12xy, exactly as shown. Option B does not produce the given square parts. Option C would give a different middle term and y² part. Option D has incorrect square pieces. Memory cue: identify the square roots of the corner squares, then add them.

Correct answer: C. x + b

The correct answer is C: x + b. The difference of squares identity is x² − b² = (x − b)(x + b). A cut-and-rearrange area model represents the same total area before and after rearrangement. If one side of the new rectangle is x − b, the remaining factor must be x + b so that the product equals x² − b². Multiplying verifies it: (x − b)(x + b) = x² + xb − xb − b² = x² − b². Option A would give (x − b)², which contains a −2xb term. Option B is the negative of the required factor and would reverse the sign of the area under ordinary positive lengths. Option D is not a factor and has the wrong degree. Memory cue: difference of squares always means difference times sum.