Visual models of identities

The correct answer is C: mn. Divide both sides of the square into segments m and n. The four parts are an m-by-m square with area m², an m-by-n rectangle with area mn, another n-by-m rectangle with area mn, and an n-by-n square with area n². Thus the term mn occurs in two separate regions. When the two rectangular areas are combined, they become mn+mn=2mn, which is the middle term in (m+n)²=m²+2mn+n². Option A appears once, in the m² corner square. Option B appears once, in the n² corner square. Option D is the combined total of the two rectangles, not the area of one repeated part. Therefore C is correct. The phrase “appears twice” asks for the individual region label, not their sum.

The correct answer is D: p²+10p+25. A square’s area is the square of its side, so the area is (p+5)². Use (u+v)²=u²+2uv+v² with u=p and v=5. This gives p²+2(p)(5)+5²=p²+10p+25. In the visual model, p² is the large p-by-p square, 5p and 5p are the two equal rectangles, and 25 is the small 5-by-5 square. Option A includes only one 5p rectangle, so it is incomplete. Option B has the correct p² and 10p but gives the wrong constant term, because 5²=25, not 5. Option C places the numbers with the wrong terms and does not follow the identity. Therefore D is correct. A quick check is to substitute p=1: the actual area is 6²=36, and D gives 1+10+25=36.

The correct answer is A: a²-2ab+b². The area of a square equals side², so the required area is (a-b)². Apply the identity (u-v)²=u²-2uv+v² with u=a and v=b. The result is a²-2ab+b². In a visual interpretation, the negative middle term reflects removing two a-by-b rectangular strips from a larger a² square, with the b² corner accounted for in the complete expression. Option B is the expansion of (a+b)²; it has the wrong sign in the middle. Option C is the difference-of-squares identity (a-b)(a+b), not the square (a-b)². Option D has neither the correct middle coefficient nor the correct structure. A common warning is that the minus sign is squared through the binomial, but the middle term does not disappear: it is -2ab.

The correct answer is B: (a+b)². Add the areas of all four parts: the first square contributes a², the two rectangles contribute ab+ab=2ab, and the second square contributes b². Therefore the total area is a²+2ab+b². By the identity (a+b)²=a²+2ab+b², this total is (a+b)². Option A includes the two corner squares but ignores both rectangles. Option C would expand to a²-2ab+b², so its middle contribution is negative, not positive. Option D is a²-b² and represents subtracting one square from another; it has no two positive ab rectangles. Thus B is the only expression matching every region in the model. Memory cue: two positive equal cross-rectangles mean +2ab, which signals the square of a sum.

The correct answer is B: the square with area b². The expression a²-b² means start with a region of area a² and subtract a region of area b². In a visual model, the larger a-by-a square represents a², and the smaller b-by-b square is the part removed. The remaining region has area a²-b², which can also be factored as (a-b)(a+b). Option A cannot be removed if we want to retain the starting a² region. Option C is only one rectangle of area ab and does not represent the subtracted square in the difference-of-squares model. Option D, 2ab, belongs to the middle contribution in square expansions such as (a+b)² or (a-b)²; it is not the part directly subtracted in a²-b². Memory cue: in “first square minus second square,” remove the second square.

The correct answer is C: r²-s². The area of a rectangle is length × breadth, so calculate (r+s)(r-s). Multiply term by term: r×r=r², r×(-s)=-rs, s×r=rs, and s×(-s)=-s². The two middle terms cancel because -rs+rs=0, leaving r²-s². This is the identity (u+v)(u-v)=u²-v². Option A incorrectly changes the minus sign before s² to plus. Option B is the expansion of (r-s)², which would require both factors to be r-s, not one sum and one difference. Option D is the expansion of (r+s)² and has a positive middle term. Therefore C is the only correct result. Memory cue: when conjugate binomials, a sum and the matching difference, are multiplied, the middle terms cancel.

The correct answer is A, 20. In the area model for (10+1)², the large square is divided into one square of area 10², one small square of area 1², and two equal rectangular regions. Each rectangle has length 10 and breadth 1, so the area of one rectangle is 10 × 1 = 10. Since there are two such rectangles, their combined area is 10 + 10 = 20, or 2 × 10 × 1 = 20. Thus, the identity is (10+1)² = 10² + 2(10)(1) + 1². Option B is only the area of one rectangle, not both. Option C is the area of the small 1-by-1 square. Option D is the area of the large 10-by-10 square. Memory cue: the middle term 2ab always comes from two equal rectangles.

The correct answer is B, 19. The phrase “square with side (20−1)” tells us to calculate the side first: 20−1 = 19. A square’s area is found by multiplying its side by itself, so its area is 19 × 19 = 19². Therefore, the area model represents the square of 19. Option A, 18, results from subtracting 2 rather than 1. Option C, 20, is the original number before the subtraction and is not the actual side. Option D, 21, would result from adding 1 instead of subtracting it. The important distinction is between the side length and the area: the side is 19, while the area is 19². Memory cue: evaluate the expression inside the brackets before deciding which number is being squared.

The correct answer is A. Imagine a large square whose side is a+b. Divide each side into parts of lengths a and b. The four resulting regions are an a-by-a square with area a², an a-by-b rectangle with area ab, a b-by-a rectangle with area ab, and a b-by-b square with area b². Adding them gives a² + ab + ab + b² = a² + 2ab + b², which is the identity (a+b)². Option B misses one of the two mixed rectangles, so it gives only ab instead of 2ab. Option C uses the wrong number and types of squares. Option D ignores the two square regions and incorrectly treats every region as an ab rectangle. Memory cue: the two cross-shaped regions are responsible for the coefficient 2 in 2ab.

The correct answer is A. Split both sides of the large square, whose side is a+b, into lengths a and b. The top-left or corresponding a-by-a region has area a². The b-by-b region has area b². The two remaining regions have dimensions a by b and b by a; each therefore has area ab. Together these two rectangles have area ab+ab = 2ab. The total area is consequently a²+2ab+b², which equals (a+b)². Option B contains only one mixed rectangle and gives a²+ab+b², so it is incomplete. Option C has four square regions instead of the required two squares and two rectangles. Option D changes two ab regions into an unrelated 2b² region. Memory cue: for a positive binomial square, count two end squares and two equal cross-rectangles.

The correct answer is D, 8. In the rectangle model, one side is split into x and 2, and the other side is split into x and 4. The constant part is the small region formed by the two numerical lengths, 2 and 4; it contains no x. Its area is therefore 2 × 4 = 8. Expanding the product confirms this: (x+2)(x+4) = x²+4x+2x+8 = x²+6x+8. Option A is only the numerical length 2, not an area. Option B is only the numerical length 4. Option C comes from adding 2+4 and is the coefficient of x in the middle term 6x, not the constant area. Option D correctly multiplies the two constant side lengths. Memory cue: in a rectangle model, the constant term is the corner formed by multiplying the two number-only parts.

The correct answer is B. Use the identity (x+y)² = x²+2xy+y². In an algebra-tile model, x² is represented by one square tile, y² by another square tile, and 2xy by two equal rectangular xy tiles. These four pieces fit together to make one large square whose side is x+y. Option A describes a rectangle with area xy, so it does not contain x², 2xy, and y². Option C corresponds to (x−y)² = x²−2xy+y²; its middle term has a minus sign, not a plus sign. Option D has area (x+y)(x−y)=x²−y², so its xy terms cancel and it does not match the given expression. Memory cue: x² plus y² with a positive 2xy middle term is the perfect-square pattern (x+y)².

The correct answer is B, 4x². In the square model of (2x+1)², the larger component square has side length 2x. The area of a square is side × side, so its area is (2x)² = (2x)(2x) = 4x². Option A, 2x², squares x but forgets to square the coefficient 2. Option C, x², would be the area of a square with side x, not side 2x. Option D, 2x, is only the side length of the larger square, not its area. The complete identity is (2x+1)² = 4x²+4x+1, where the other pieces are two rectangles of area 2x each and a 1-by-1 square. Memory cue: when a whole term is the side of a square, every factor in that term must be squared: (2x)² = 2²x² = 4x².

The correct answer is B. The total area of a figure made from non-overlapping regions is found by adding the areas of all its parts. Here the parts have areas a², ab, ab and b². Therefore the total is a²+ab+ab+b² = a²+2ab+b². This is also the expansion of (a+b)², so the model represents the square of a+b. Option A subtracts the two ab regions, even though they are visible positive regions that must be added. Option C contains only a² and b² and omits both rectangular regions. Option D doubles the square regions and does not include the required ab parts. The coefficient 2 appears because there are two equal rectangles, not because either rectangle has area 2ab by itself. Memory cue: count every separate region and add its area; two ab regions produce +2ab.

The correct answer is B, (a+b)². A square with side a+b can be divided in both directions into lengths a and b. This creates one a² square, one b² square, and two equal rectangles, each with dimensions a and b and therefore area ab. The two rectangles together have area ab+ab=2ab, giving (a+b)²=a²+2ab+b². Option A, (a−b)², has the expansion a²−2ab+b², so its mixed contribution is negative in the identity. Option C is only the sum of two square areas and has no two rectangular cross-parts. Option D, a²−b², represents a difference of squares and also does not contain the 2ab middle term. Memory cue: a plus sign between a and b produces +2ab; a minus sign in a squared binomial produces −2ab.

The correct answer is C, 100. In the area model for (10+5)², the large square has side length 10, because the first part of each side is 10. The area of this large square is therefore 10 × 10 = 10² = 100. The complete model also contains a smaller 5-by-5 square with area 25 and two 10-by-5 rectangles, each with area 50. Option A, 25, is the area of the small square, not the large one. Option B, 50, is the area of one rectangle, not the large square. Option D, 150, is not the area of any single component in this model and comes from confusing the pieces. The total area is 225, which is 15², but the question asks only for the large square. Memory cue: identify the requested region first; the large square uses side 10, so calculate 10².

The direct answer is option C, 9. Imagine drawing two internal lines parallel to each side of the square. In one direction, the square is divided into 3 strips, and in the perpendicular direction it is also divided into 3 strips. Each of the 3 rows intersects each of the 3 columns, so the small regions must be counted by multiplication: 3 × 3 = 9. Option A, 3, counts only the rows or only the columns, not all enclosed regions. Option B, 6, adds the number of rows and columns, but grid regions are formed by their intersections, so addition is unsuitable. Option D, 12, has no valid counting basis for this 3-by-3 grid. The symbols a, b and c describe lengths and may affect the sizes of the regions, but they do not change the number of regions. Memory cue: number of grid regions = number of rows × number of columns.

The direct answer is option B: square parts. In the area model, a large square has side length a+b+c. The horizontal and vertical divisions create smaller regions. The region with side a in both directions has area a × a = a^2, so it is a square. Similarly, the region with side b in both directions has area b × b = b^2, and the region with side c in both directions has area c × c = c^2. Option A is wrong because a rectangle has two possibly different side lengths, such as ab or ac; these mixed products occur in the model but are not the pure square terms. Option C describes a boundary line, not a two-dimensional area. Option D refers to the outside boundary, whereas a^2, b^2 and c^2 are interior areas. Memory cue: a variable multiplied by itself gives the area of a square whose side is that variable.

The direct answer is option A: ba is equal to ab. In the visual model, an ab rectangle has one side of length a and the other of length b, so its area is a × b = ab. If the orientation is reversed, the same rectangle can be described as having side b first and side a second. Its area is then b × a = ba. Multiplication is commutative, which means changing the order of the factors does not change the product: ab = ba. Option B would be the area of a square with both sides a. Option C would be the area of a square with both sides b. Option D would involve side lengths c and c, not a and b. The two ab-type rectangles in the model may look differently placed, but they have equal areas. Memory cue: turning a rectangle around changes its orientation, not its area.

The direct answer is option A: (x+2) and (x+3). In an area model, split one side into x and 2, and split the other side into x and 3. Multiplying every pair of pieces gives four regions: x·x = x^2, x·3 = 3x, 2·x = 2x, and 2·3 = 6. These are exactly the four areas stated in the question. Therefore the complete rectangle has dimensions (x+2) and (x+3), and its total area is (x+2)(x+3) = x^2+5x+6. Option B would produce x^2+7x+6, not the given terms. Option C would produce x^2+8x+12, so both middle and constant parts differ. Option D would produce x^2+6x+9. A useful check is that the constant region must be the product of the two numerical parts: 2 × 3 = 6.

The direct answer is option B, 4. Write 18 as 20−2. The identity (p−q)^2 = p^2−2pq+q^2 shows that (20−2)^2 = 20^2−2(20)(2)+2^2. In the visual area model, begin with a square of side 20. Removing a strip of width 2 from one side and another strip of width 2 from the other side creates overlap at the corner. That small corner is a square with side 2, so its area is 2 × 2 = 4. It must be added back because the overlap was subtracted twice. Option A, 2, is only the side length of the small square, not its area. Option C, 40, is the product 20 × 2 associated with one rectangular part. Option D, 80, is twice that product and belongs to the middle term 2pq. Memory cue: in (p−q)^2, the final small-square area is q^2, not q.

The direct answer is option C, 7x. Use the rectangle area model by multiplying the two side expressions: (x+5)(x+2). Each part of the first side is multiplied by each part of the second side, giving x·x = x^2, x·2 = 2x, 5·x = 5x, and 5·2 = 10. Thus the complete expansion is x^2+2x+5x+10. The x-terms are the two like terms 2x and 5x, so their total is 2x+5x = 7x. Option A includes only the first x-term. Option B includes only the second x-term. Option D, 10x, incorrectly treats the constant product 10 as an x-term or multiplies coefficients without following the model. The term x^2 is quadratic and the term 10 is constant, so neither belongs in the requested total. Memory cue: collect only terms with the same variable and the same exponent; add their coefficients.

The direct answer is option A. Start with one large square whose side is a+b. Divide each side into pieces of lengths a and b. The four resulting regions are an a-by-a square with area a^2, an a-by-b rectangle with area ab, a b-by-a rectangle with area ab, and a b-by-b square with area b^2. Adding their areas gives a^2+ab+ab+b^2 = a^2+2ab+b^2, which equals (a+b)^2. Option B describes a rectangle with area (a+b)(a−b) = a^2−b^2, so it represents a difference of squares, not a square of a sum. Option C lists separate shapes and misses the two cross-rectangles required by the identity. Option D involves subtraction and does not create the four positive parts in the stated expansion. Memory cue: the middle term 2ab means there must be two congruent ab rectangles.

The direct answer is option B, x+6. Compare the expression with the identity (p+q)^2 = p^2+2pq+q^2. Here p=x and q=6: x^2+2(x)(6)+6^2 = x^2+12x+36. Therefore x^2+12x+36 = (x+6)^2, so the side of the square is x+6. In the area model, the x^2 region is the large x-by-x square, the two cross-regions each have area 6x, and the small corner square has area 36 and side 6. Option A, x+12, would produce a middle term 24x and a final term 144. Option C, x+3, would produce 6x and 9. Option D would produce negative middle term −12x, so it belongs to a different pattern. A safe check is to halve the coefficient of x: 12 ÷ 2 = 6, and then verify that 6^2 = 36.

The direct answer is option A: a^2+ab+ab+b^2. A square with side a+b is divided by one line in each direction, producing four regions. The a-by-a square has area a^2. The a-by-b and b-by-a rectangles each have area ab, so together they contribute ab+ab. The b-by-b square has area b^2. Thus the total area is a^2+ab+ab+b^2 = a^2+2ab+b^2 = (a+b)^2. Option B includes only the two square regions and omits both cross-rectangles. Option C includes only one ab rectangle, so it misses half of the middle contribution. Option D has no correct area-model decomposition: it replaces the two ab rectangles with 2b^2, which is a different type of area. The visual model is especially useful because it shows why the middle term has coefficient 2. Memory cue: a binomial square always has two cross-products.