Visual models of identities

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. 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.