Visual models of identities

The direct answer is A: two rectangles, each with area ab. Imagine a square of side a + b divided into four parts. One part is an a by a square with area a². Another is a b by b square with area b². The remaining two parts are rectangles with dimensions a and b, so each has area ab. Adding both equal rectangles gives ab + ab = 2ab. Option A is correct because the middle term is exactly the combined area of these two rectangles. Option B contributes only a², the first term. Option C contributes only b², the last term. Option D concerns boundary length, not area, and cannot produce the term 2ab. Memory cue: in (a+b)², the two mixed rectangles create the middle term.

The direct answer is B. The area of a rectangle is length × breadth, so its area is (x + 3)(x − 3). This is the sum-and-difference identity (a+b)(a−b) = a² − b², with a = x and b = 3. Therefore the area is x² − 3² = x² − 9. Option A expands (x+3)², which would describe a square whose both sides are x+3. Option B correctly matches the given rectangle and the difference-of-squares identity. Option C expands (x−3)², another square not shown here. Option D has the wrong sign: (x+3)(x−3) equals x²−9, not x²+9. Memory cue: opposite signs in two factors usually signal a difference of squares.

The correct answer is C: each rectangle has area uv. Start with the basic rule that the area of a rectangle equals length × breadth. In the square model, one side of each smaller rectangle has length u and the other has length v. Therefore, the area of one rectangle is u × v = uv. There are two such rectangles, so together they contribute uv + uv = 2uv; however, the question asks for each rectangle separately, so the answer is only uv. Option A, (u+v)², is the area of the entire large square, not one rectangle. Option B, u²v², incorrectly multiplies the areas of the two square pieces and also has the wrong dimensions. Option D, 2uv, is the combined area of both rectangles, not the area of one. Memory cue: one rectangle gives uv; two equal rectangles give 2uv.

The correct answer is A: x+3. To find the side of a square from its area, look for a perfect-square expression. Use the identity (p+q)² = p²+2pq+q². Here x² is p², 9 is 3², and the middle term is 6x = 2×x×3. Therefore x²+6x+9 = x²+2×x×3+3² = (x+3)². Since area of a square equals side², its side is x+3. Option B would give (x+6)² = x²+12x+36, not the given expression. Option C gives (x−3)² = x²−6x+9, whose middle term has the wrong sign. Option D gives (3x+1)², which has a leading term 9x². Memory cue: half the middle-term coefficient, 6/2=3, gives the constant in the side.

The correct answer is B: y−4. Recognize the perfect-square pattern p²−2pq+q² = (p−q)². In the expression y²−8y+16, the first term is y², the last term is 16=4², and the middle term is −8y = −2×y×4. Therefore y²−8y+16 = (y−4)². A square with side y−4 has exactly this area. Option A gives (y+4)² = y²+8y+16, so its middle term has the opposite sign. Option C gives (y−8)² = y²−16y+64, which has incorrect middle and constant terms. Option D gives a leading term 16y² when squared, so it cannot match. Memory cue: a negative middle term in a perfect square means the side uses a minus sign.

The correct answer is D: 2ab. Each of the two rectangles has one side a and the other side b. Therefore, the area of one rectangle is a×b=ab. Since there are two equal rectangles, their combined area is ab+ab=2ab. This explains the identity (a+b)² = a²+2ab+b²: the two square tiles contribute a² and b², while the two rectangular tiles provide the middle term 2ab. Option A is the difference of two squares and does not represent the two positive rectangle areas. Option B is the area of only one rectangle. Option C is the sum of the two square areas, not the rectangles. Memory cue: the middle term in a square expansion comes from the two cross-rectangles, so it is twice ab.

The correct answer is A: 42 square centimetres. In the identity (a+b)², the two equal rectangles together have area 2ab. Here a=7 and b=3, so their combined area is 2×7×3 = 14×3 = 42 cm². You can also see the two separate rectangles: each has area 7×3=21 cm², and 21+21=42 cm². Option B is the area of only one rectangle, not both. Option C does not follow from the rectangle-area calculation and is not a component of the square decomposition. Option D is the area of the entire 10 cm by 10 cm square, 10²=100 cm², not just the two rectangles. The remaining two square parts have areas 49 cm² and 9 cm², and 49+42+9=100, confirming the model. Memory cue: two cross-parts always give 2ab.

The correct answer is A, 119 square units. The area of a square is found by multiplying its side by itself, or using side². The large square has area 12² = 144 square units. The removed small square has area 5² = 25 square units. Because the question asks for the area left after removal, subtract the smaller area from the larger area: 144 − 25 = 119 square units. Option A is correct because it represents the remaining region. Option B is only the area of the original large square, before anything is removed. Option C is only the area of the removed square. Option D is 13² and does not describe either square or their difference. Memory cue: area difference means large square area minus small square area.

The correct answer is B, n² − 4n + 4. A square’s area is the product of its side by itself, so the area is (n − 2)(n − 2), or (n − 2)². Apply the identity (a − b)² = a² − 2ab + b². Here a = n and b = 2, so (n − 2)² = n² − 2(n)(2) + 2² = n² − 4n + 4. Option A has the wrong sign in the middle term and corresponds to (n + 2)². Option B correctly includes the square of the first term, the negative cross-term, and the square of 2. Option C incorrectly uses −2n instead of −4n. Option D is n² − 4, which is a difference of squares, not the square of a difference. Memory cue: when a binomial with a minus sign is squared, the middle term is negative and must be doubled.

Direct answer: Option B, two, is correct. Divide the horizontal side into lengths a and b, and do the same vertically. The resulting four regions are an a² square, an ab rectangle, another ab rectangle, and a b² square. The two cross-regions each have dimensions a by b, so each has area ab. Together they contribute 2ab in the identity (a+b)²=a²+2ab+b². Option A counts only one cross-region and overlooks its matching region. Option C is too many because there are only two mixed-length regions. Option D confuses the total number of regions, which is four, with the number whose area is ab. Memory cue: the coefficient 2 in 2ab tells you there are two ab rectangles.

Direct answer: Option A is correct. Since 16=4², the expression x²−16 means the area of a square of side x minus the area of a square of side 4. This is the visual meaning of a difference of two squares: x²−4²=(x+4)(x−4). Option B represents x²+16, not subtraction. Option C gives the product 16x, which is a different expression and does not describe subtracting one square area from another. Option D gives 4x+4x=8x, also unrelated to x²−16. The important step is to recognise the constant 16 as 4². Memory cue: a minus sign between two square areas means remove the smaller square from the larger one.

The direct answer is B, m²+10m+16. The area of the rectangle is the product (m+2)(m+8). Split the sides into m and 2, and m and 8. The four area regions are m·m=m², m·8=8m, 2·m=2m, and 2·8=16. Adding them gives m²+8m+2m+16=m²+10m+16. Option A is wrong because 16 is the product of the constants and must be a constant term; it cannot become 16m. Option C incorrectly combines the variable terms as 8m−2m instead of 8m+2m. Option D has a negative constant, but both constant side lengths are positive, so 2·8=+16. The area model is simply the distributive property shown as four smaller rectangles. A useful rule is (m+r)(m+s)=m²+(r+s)m+rs; here r=2 and s=8, so the result is m²+10m+16.

The direct answer is A: both areas are equal. A visual proof works by calculating the area of one unchanged figure in two different ways. The large square has side a+b, so its area is (a+b)^2. If the same square is divided into one a-by-a square, two a-by-b rectangles, and one b-by-b square, the parts have areas a², ab, ab, and b². Their sum is a²+ab+ab+b²=a²+2ab+b². Because both calculations describe the same complete square, the expressions must be equal. B is false because neither expression is generally larger. C is false because the second expression is an area, not always zero. D is false because the figure is the same, only the partition or calculation differs. Memory cue: same figure plus two valid area calculations gives an identity.

The direct answer is B: a square with side (x−3). Compare the expression with the identity (p−q)^2=p²−2pq+q². The first term x² is x², the last term 9 is 3², and the middle term is −6x=−2(x)(3). Therefore x²−6x+9=(x−3)^2. A would expand to x²+6x+9, so its middle sign is wrong. C would produce x²−12x+36, not the given expression. D would produce x²+18x+81, so both its middle and constant terms are wrong. In a visual model, the negative middle term means the side pieces are subtracted rather than added. Memory cue: a negative middle term usually signals a difference square.

The direct answer is A: 119. Substitute x=12 into the two side lengths of the visual rectangle. The first side is x+5=12+5=17, and the second side is x−5=12−5=7. Rectangle area equals length times width, so the area is 17×7=119. The same result comes from the difference-of-squares identity x²−25=x²−5²=(x+5)(x−5): 12²−25=144−25=119. B is only x² and forgets the subtraction. C is 13² and does not represent the given expression. D is 10² and does not use the correct two side lengths. Memory cue: when a difference of squares is already factored, substitute into both factors before multiplying.

The direct answer is A: the constant corner is positive 21. In the rectangle model, the constant corner comes from multiplying the two constant side pieces, −3 and −7. Their product is (−3)(−7)=+21 because the product of two negative numbers is positive. Expanding the whole expression confirms this: (x−3)(x−7)=x²−7x−3x+21=x²−10x+21. B incorrectly keeps a negative sign after multiplying two negatives. C and D use 10x, which belongs to the combined x-terms, not to the constant corner. The constant corner contains no x because both factors used there are constants. Memory cue: same signs multiply to a positive result; different signs multiply to a negative result.

The direct answer is D. When both sides of a square of side a+b are divided into lengths a and b, the partition contains an a by a square, two a by b rectangles, and a b by b square. Each of the two rectangles has the same pair of side lengths, a and b, so each area is a×b=ab. Their positions may be different, or one may look rotated, but rotation does not change area. A is wrong because the large square has area (a+b)², not 2ab. B is wrong because equal dimensions give equal areas. C is wrong because the small square is b², not ab. The useful cue is: same length and breadth means same rectangle area.

The direct answer is D: the difference is 1. Write the comparison as (x²+8x+16)−(x²+8x+15). The common parts x² and 8x cancel, leaving 16−15=1. In an area model, both figures contain the same large x² region and the same combined strip area 8x. Only the corner pieces differ: one has area 16 and the other has area 15. Therefore the second model is larger by one square unit. A is wrong because 16 is the second corner itself, not the excess over 15. B and C are common strips present in both models, so they cannot create the difference. Memory cue: when comparing expressions, cancel identical terms before interpreting the remaining piece.

The correct answer is A: (101-99)(101+99). The difference-of-squares identity is P^2-Q^2=(P-Q)(P+Q). Taking P=101 and Q=99 gives 101^2-99^2=(101-99)(101+99)=2×200=400. In a visual comparison, the larger square and the smaller square overlap; the remaining strips can be rearranged into a rectangle whose sides are the difference and the sum of the original side lengths. Option A represents exactly this rectangle. Option B calculates only the square of the difference, 2^2=4, which is not the difference of the two large square areas. Option C adds the areas instead of subtracting them. Option D is the product of the side lengths and is not equal to the difference of their squares. A useful mental shortcut is to centre the numbers around 100: 101^2-99^2=(100+1)^2-(100-1)^2=4×100×1=400. Memory cue: difference of nearby squares is often much faster as difference times sum.

The direct answer is B: 14x. The complete square has side x+7, so its area is (x+7)². In the model, the x-by-x part has area x², and the corner square has area 49 = 7². The two remaining rectangles are equal: each has length x and width 7, so each area is 7x. Adding them gives 7x + 7x = 14x. The algebra confirms this because (x+7)² = x² + 14x + 49; after removing x² and 49, the remaining area is 14x. Option A is only the area of one rectangle. Option C incorrectly multiplies quantities and does not describe either the strips or their sum. Option D adds the two square regions instead of the rectangular regions. Remember that the coefficient 2 in 2·x·7 represents the two equal rectangles.

The direct answer is C: 2. Divide each side of the square of side p+q into parts p and q. The square then contains four regions: a p-by-p square with area p², a q-by-q square with area q², and two rectangles, each measuring p by q and therefore having area pq. These two rectangles lie in opposite, or diagonal, positions. Option A counts only one of them and misses its matching rectangle. Option B has no basis in the four-part decomposition. Option D counts all four regions, not only the rectangles with area pq. The identity (p+q)² = p² + 2pq + q² gives the same result: the coefficient 2 before pq tells us that there are two pq rectangles. A useful memory cue is that the middle term in a square of a sum comes from two equal cross-rectangles.

The direct answer is A: 16y² − 40y + 25. Recognise the first and last terms as squares: 16y² = (4y)² and 25 = 5². If the two strips are subtracted, the visual identity is (4y−5)². Expanding gives (4y−5)² = (4y)² − 2(4y)(5) + 5² = 16y² − 40y + 25. The middle term is negative because the strips are being removed, and its coefficient is 2 because there are two equal strips. Option B has the correct square terms but a positive middle term, which would represent adding strips. Option C incorrectly squares 4y as 4y² instead of 16y². Option D uses only one product contribution, so it gives −20y rather than −40y. Memory cue: in (A−B)², the middle term is −2AB.

The direct answer is A: the small corner has side 6 and the large part has side 5a. In a square-area model, a square with area 25a² has side √(25a²)=5a, while a square with area 36 has side √36=6. The middle term confirms the arrangement: two rectangles each have area (5a)(6)=30a, giving 60a altogether. Thus the expression is (5a+6)² = 25a²+60a+36. Option B lists the same two lengths in the reverse order, but the question asks for the small corner first. Option C gives area values rather than side lengths, and 25a is not the side obtained from 25a². Option D does not square to the required first and last terms. The key habit is to take square roots of the square tiles and then check the two cross-rectangles.

Correct answer: C, a combined strip area of 24x. The square has side x+12, so its total area is (x+12)^2=x^2+2(x)(12)+12^2=x^2+24x+144. The x^2 region is already present, and the corner of area 144 is present because 12^2=144. The remaining two rectangles each have dimensions x and 12, so each has area 12x. Together they contribute 12x+12x=24x. Option A gives only one of the two equal strips and leaves the model incomplete. Option B incorrectly multiplies 12 and 12x and has the wrong degree and coefficient. Option D repeats existing regions rather than supplying the missing middle part. Memory cue: for (p+q)^2, the two side rectangles always combine to 2pq.

Correct answer: A, (x+y)^2=x^2+2xy+y^2. The large figure is a square whose side is made by joining lengths x and y, so its side length is x+y and its total area is (x+y)^2. The four parts have areas x^2, xy, xy, and y^2. Adding them gives x^2+xy+xy+y^2=x^2+2xy+y^2. Option B belongs to a difference square and would require negative middle rectangles. Option C is the difference-of-squares identity, which has one positive and one negative square rather than two xy rectangles. Option D is false because (x+y)(x-y)=x^2-y^2, not x^2+y^2. Memory cue: two positive xy rectangles signal the plus-square identity.