Visual models of identities

The correct answer is B: 2xz. Expand the square of a sum of three terms: (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz. The square terms come from pairing each term with itself. The mixed terms come from pairing two different terms; each pair appears twice because xy and yx represent equal areas, and similarly for xz and zx or yz and zy. The question already lists x^2, y^2, z^2, 2xy, and 2yz, so the missing pair is x and z, contributing 2xz. Option A gives only one xz rectangle and misses its equal partner. Option C is a side length, not an area term. Option D has three factors and is cubic, whereas the square of a sum produces degree-two terms. Memory cue: with three terms, list all pairs: xy, xz, yz, and double each one.

Direct answer: Option A, 1. A perfect-square expression has the form (x+a)²=x²+2ax+a². For the first expression, x²+6x+9=x²+2·x·3+3²=(x+3)², so its square has side x+3. For the second expression, x²+8x+16=x²+2·x·4+4²=(x+4)², so its side is x+4. The second side minus the first side is (x+4)−(x+3)=1. Option A is correct. Option B, 2, wrongly compares the coefficients 6 and 8 directly. Option C, 3, is the small constant in the first factor, not the difference of the sides. Option D, 4, is the small constant in the second factor. Remember: factor each perfect square first; compare side expressions, not their areas or middle coefficients.

Direct answer: Option B, 6x. Rewrite the expression as (2x+3)²: (2x)²+2(2x)(3)+3²=4x²+12x+9. In the area model, the two middle rectangles are equal. Their combined area is the middle term 12x, so each rectangle has area 12x÷2=6x. Option A, 3x, is only half of the constant-side length multiplied by x and does not give the required total when doubled. Option B is correct because 6x+6x=12x. Option C, 12x, is the area of both strips together, not each one. Option D, 9x, confuses the constant square’s area 9 with a strip area. A useful check is to identify the two equal cross-products: each is (2x)(3)=6x.

Direct answer: Option C, 5. Write the expression as x²−10x+25=x²−2·x·5+5²=(x−5)². In the area model, the corner square is the constant square whose area is 25. Since the area of a square is side², its side is √25=5. The negative middle term indicates subtractive strips; it does not make the corner side negative. Option A, 25, is the area of the corner square, not its side. Option B, 10, is the coefficient of the middle term and equals 2×5, not the side itself. Option C is correct because 5²=25 and 2x×5=10x. Option D, 15, neither has square 25 nor produces the required middle term. Memory cue: in x²−2kx+k², the small corner square has side k.

Direct answer: Option B, they cancel to zero. Multiply the side expressions: (x+a)(x−a)=x·x−ax+ax−a²=x²−a². In the visual rectangle, one middle strip has area ax and the other has area −ax. These are equal in magnitude but opposite in sign, so their sum is ax+(−ax)=0. Option A is wrong because adding two positive ax terms would apply to (x+a)(x+a), not to a sum-and-difference product. Option B is correct. Option C is wrong because a² comes from the product of the two a-parts and is a final constant term, not the combined linear strips. Option D is wrong because the product is x²−a², not x²+a². Memory cue: (x+a)(x−a) has no middle term—the plus and minus cross-products cancel.

Direct answer: Option A, because the identity is based on area. The expression x²+2xy+y² represents the area of a square with side x+y: one x² square, two xy rectangles, and one y² square. Adding these four areas gives (x+y)². Perimeter measures boundary length and has different units from area, so it cannot replace the sum of the tile areas. Option A is correct. Option B is false: perimeter is not always x²; for a square of side x+y it is 4(x+y). Option C is false because every rectangle has an area, found by length×width. Option D is also false: y² is an area component, not something to add to a perimeter. Common warning: do not mix area and boundary length, even when the same diagram contains both.

Direct answer: Option B, 8x. The required difference is (x+2)²−(x−2)². Expand carefully: (x+2)²=x²+4x+4 and (x−2)²=x²−4x+4. Subtracting gives x²+4x+4−x²+4x−4=8x. Thus the four visual strips together have area 8x. Option A, 4x, is only one of the two opposing linear contributions after the comparison, not the full difference. Option B is correct. Option C, x²−4, equals (x+2)(x−2), a product of the two side expressions, not the difference of the square areas. Option D, 16, is the constant obtained from 4·4 in the shortcut but omits x. The shortcut (a+b)²−(a−b)²=4ab gives 4·x·2=8x immediately.

Direct answer: Option A, 7x and 8x. To split the middle term in a rectangle model, find two numbers whose sum is 15 and whose product is 56. The numbers 7 and 8 satisfy both conditions: 7+8=15 and 7×8=56. Therefore x²+15x+56=(x+7)(x+8), and the two linear strips have areas 7x and 8x. Option A is correct. Option B gives 6+9=15, but 6×9=54, not 56. Option C gives 4+11=15, but 4×11=44. Option D gives 5+10=15, but 5×10=50. The memory cue is to check both the sum and the product; matching only the middle coefficient is not enough.

Direct answer: Option A, (2a−b) and (2a+b). Recognise a difference of squares: 4a²−b²=(2a)²−b². Using p²−q²=(p−q)(p+q), with p=2a and q=b, the factorisation is (2a−b)(2a+b). In a visual model, rearranging the positive and negative square tiles forms a rectangle with these two side lengths. Option A is correct, and multiplication verifies it: (2a−b)(2a+b)=4a²−b². Option B produces 16a²−b², so its first term is too large. Option C produces (2a−b)²=4a²−4ab+b², with an unwanted middle term and the wrong sign of b². Option D produces a²−b², missing the factor 4 in 4a². Memory cue: first rewrite 4a² as (2a)² before applying the difference-of-squares identity.

Direct answer: Option A, x²+81. Decompose the expression as x²+18x+81=x²+9x+9x+81. In the square model, x² is one square, 81 is the other square, and the two middle strips together have area 18x. Removing the strips leaves x²+81. Option A is correct. Option B, 18x, is precisely the area removed, not the area that remains. Option C incorrectly keeps one strip term together with x² and ignores the separate constant square. Option D confuses 81=9² with 9; the constant square’s area is 81, not 9. Another check is (x+9)²=x²+18x+81, so the two square components are x² and 9²=81. Keep square-tile areas and rectangular-strip areas separate.

Direct answer: Option A, two xy rectangles. The identity (x+y)²=x²+2xy+y² comes from partitioning a large square into four parts: one square of area x², two rectangles each of area xy, and one square of area y². The two rectangles contribute xy+xy=2xy. If the expression is incorrectly reduced to x²+y², exactly this cross-term contribution has been omitted. Option A is correct. Option B is wrong because x² is still present in the incorrect expression. Option C is wrong because y² is also still present. Option D is wrong because the outer boundary is not the missing algebraic area term; the missing part is inside the model. Common warning: (x+y)² is not x²+y²; always include the two cross rectangles, or the 2xy term.

The correct answer is B. Imagine dividing each side of the large square into two lengths, a and b. These divisions create four regions: one square with side a and area a², one square with side b and area b², and two rectangles. Each rectangle has length a and width b, so each has area ab. Therefore, the complete area is a² + ab + ab + b² = a² + 2ab + b². Option A is wrong because there is only one a² square. Option C is wrong because the b² corner square is present. Option D is wrong because it omits one ab rectangle and therefore misses the middle term 2ab. Memory cue: the square of a sum has two cross-products: (a + b)² = a² + 2ab + b².

The correct answer is B, 12y. The side y + 6 is divided into a part of length y and an extra part of length 6. In the area model, the large y by y square has area y². The two remaining strips are each formed by multiplying y by 6, so each strip has area 6y. Adding the equal strips gives 6y + 6y = 12y. Option A gives the area of only one strip, not both. Option C is incorrect because 36 would be the area of a 6 by 6 corner, and writing 36y has the wrong dimensions for that corner. Option D adds the y² square and the 6 by 6 corner, but the question asks only for the two strips. Common confusion: equal strips must be counted twice; this is why the middle term in (y + 6)² is 12y.

The correct answer is C. The full square has side a + 5, so its area is (a + 5)². In the visual model, one large square has area a². There are two rectangular strips, each measuring a by 5, so their areas are 5a and 5a. The small corner is a 5 by 5 square, giving 25. Adding all four regions gives a² + 5a + 5a + 25 = a² + 10a + 25. Option A counts only one strip and therefore has 5a instead of 10a. Option B gives an incorrect coefficient for a and an incorrect constant arrangement. Option D has the correct middle coefficient but the wrong constant, 5 instead of 25. The identity (a + b)² = a² + 2ab + b² confirms the result when b = 5.

The correct answer is D. Start with the large square of side m and area m². Because the required side is m − 3, one strip of dimensions m by 3 is removed horizontally and another strip of dimensions m by 3 is removed vertically. Their total removed area is 3m + 3m = 6m. The small 3 by 3 corner has been removed twice, so it must be added back once; its area is 9. Thus the final area is m² − 6m + 9, which is also the identity (m − 3)². Option A subtracts only 3m and misses one strip. Option B uses the signs for a sum rather than a difference. Option C has incorrect coefficients and does not result from the two equal strips. Memory cue: for (u − v)², use u² − 2uv + v²; the final corner is added back.

The correct answer is A. The area of a rectangle is length multiplied by width, so calculate (p + 4)(p + 7). The area model divides this product into four parts: p × p = p², p × 7 = 7p, 4 × p = 4p, and 4 × 7 = 28. Combining like terms gives p² + 7p + 4p + 28 = p² + 11p + 28. Option B places 28 with p and 11 as the constant, reversing the roles of the products. Option C uses 3p, which comes from subtracting 7 and 4 rather than adding the two cross-products. Option D has the correct middle coefficient but the wrong constant; 4 × 7 is 28, not 11. A safe method is to multiply every part of the first binomial by every part of the second.

The correct answer is B. The rectangle’s area is (r + 8)(r − 8). This is the difference-of-squares pattern (u + v)(u − v) = u² − v², with u = r and v = 8. Therefore the area is r² − 8² = r² − 64. If expanded directly, the products are r² − 8r + 8r − 64; the two middle terms cancel, leaving r² − 64. Option A has the wrong sign before 64. Options C and D incorrectly retain a middle term of ±16r, but the two cross-products are opposites and cancel. The visual model shows two equal cross-regions with opposite signs, so no r term remains. Memory cue: conjugate binomials produce a difference of squares, not a sum.

The correct answer is C. In an area grid, the x² part must come from x × x. The 9x part can come from x × 9, and the 2x part can come from 2 × x. The remaining corner must then be 9 × 2 = 18. Therefore the two side lengths are x + 9 and x + 2, because (x + 9)(x + 2) = x² + 2x + 9x + 18. Option A would create different cross-products and a corner of 198, not 18. Option B would produce two 9x strips and a 81 corner. Option D would make an x × 18 strip and a corner 36, so it does not match the given regions. Memory cue: identify the two strip widths from the coefficients of x, then check that their product equals the constant corner.

The correct answer is B: 4x² + 20x + 25. The square’s area is (2x + 5)². Use (a + b)² = a² + 2ab + b² with a = 2x and b = 5. The large square has area (2x)² = 4x². There are two cross-rectangles, each with area (2x)(5) = 10x, so together they contribute 20x. The small corner is 5² = 25. Adding the regions gives 4x² + 20x + 25. Option A counts only one cross-rectangle, so its middle term is too small. Option C incorrectly treats (2x)² as 2x² instead of 4x². Option D has incorrect coefficients for the middle and constant terms. Always square the coefficient as well as the variable: (2x)² = 4x².

The correct answer is A. Divide each side of the large square into lengths a and b. The resulting model contains an a by a square with area a², a b by b square with area b², and two cross-rectangles. Each cross-rectangle has sides a and b, so each area is ab. Since there are two such rectangles, their combined area is ab + ab = 2ab. Option B represents only a², not 2ab. Option C represents only b². Option D represents two a² squares, giving 2a², not 2ab. The coefficient 2 comes from the number of identical rectangles, while the product ab comes from their side lengths. Memory cue: in (a + b)², the middle term is made by two equal cross-rectangles.

The correct answer is A. The large square has side length x + y, because its side is made from an x segment and a y segment. Its four regions have areas x², xy, xy, and y². Adding them gives x² + xy + xy + y² = x² + 2xy + y². Since the whole figure is a square of side x + y, its area is also (x + y)². Hence (x + y)² = x² + 2xy + y². Option B has negative middle terms, which would correspond to a subtraction model. Option C is a valid difference-of-squares identity, but it does not match the four positive regions shown. Option D is false because the product of conjugates gives x² − y², not x² + y². Check the signs of the regions before choosing an identity.

The correct answer is A: \(64\). In an area model, first recognise the perfect-square identity \((a+b)^2=a^2+2ab+b^2\). Here, \(z^2+16z+64\) can be written as \(z^2+2(z)(8)+8^2=(z+8)^2\). Therefore, the large square has side \(z+8\). The model contains a \(z\)-by-\(z\) square with area \(z^2\), two rectangles whose combined area is \(8z+8z=16z\), and a small corner square with side 8. Its area is \(8\times8=64\). Option A is correct because the constant term is the part made only from the fixed length 8. Option B is the combined area of the two variable rectangles, not the corner. Option C is only one possible rectangle area, and it is not the corner square. Option D is the large variable square. A useful memory cue is: in \((z+8)^2\), the last term \(8^2\) gives the fixed corner area.

The correct answer is B: \(10n\). Begin with a large square of side \(n\), so its area is \(n^2\). To make a square of side \(n-5\), a strip of width 5 is removed along one direction and another strip of width 5 is removed along the other direction. In the usual visual identity model, each full strip is treated as having area \(n\times5=5n\). Therefore, the two strips together have area \(5n+5n=10n\). This is also visible from the identity \((n-5)^2=n^2-10n+25\): the term \(-10n\) records the combined removal of the two strips, while the \(+25\) term accounts for the small corner square that is restored or handled separately after the overlap. Option A gives only one strip. Option C uses an incorrect product and does not represent either the two strips or the corner. Option D is the area of the 5-by-5 corner square, not the total strip area. A common confusion is to mix the overlap with the strips; the question asks for the two strip contributions, represented by \(10n\).

The correct answer is B: \((t+3)\) and \((t+4)\). To arrange the area tiles into a rectangle, find two numbers whose product is the constant term 12 and whose sum is the coefficient of \(t\), namely 7. The numbers 3 and 4 satisfy both conditions: \(3\times4=12\) and \(3+4=7\). Therefore, \(t^2+7t+12=(t+3)(t+4)\). On multiplying, the four regions are \(t^2\), \(4t\), \(3t\), and 12; their sum is \(t^2+7t+12\). Option A has factors 1 and 12, whose sum is 13, so its middle term would be \(13t\). Option C has the correct product 12 but gives \(2+6=8\), producing \(8t\). Option D gives both an incorrect product and an incorrect sum for this trinomial. The key check is always both product and sum, not just the constant term.

Answer: A, \(24x\). In the visual model, split each side of the square into lengths \(4x\) and \(3\). This creates four regions: a square of area \((4x)^2=16x^2\), a square of area \(3^2=9\), and two mixed rectangles. Each mixed rectangle has length \(4x\) and width \(3\), so its area is \(4x\times3=12x\). Since there are two such rectangles, their combined area is \(12x+12x=24x\), or directly \(2ab=2(4x)(3)=24x\). Option A is correct. Option B gives the area of only one mixed strip, not both. Option C is the large \(4x\)-square, not a mixed region. Option D is the small constant square. Memory cue: in \((a+b)^2\), the two mixed parts together always contribute \(2ab\).