Visual models of identities

Answer: B, (2x + 3)(2x + 5). A rectangle model for a product has an x-by-x part, two cross parts, and a constant corner. Multiplying the two proposed dimensions gives (2x + 3)(2x + 5) = 4x² + 10x + 6x + 15 = 4x² + 16x + 15. The two cross rectangles are therefore 10x and 6x, and their corner is 3×5 = 15, exactly as required. Option A has cross terms 6x and 15x, although its corner is 15. Option C has a different structure and expands to 6x² + 100x + 150, so it does not match. Option D gives x² + 16x + 60, with corner 60 rather than 15. Common confusion: the two cross terms come from opposite pairings, so each must be checked separately; checking only the constant corner is insufficient.

Answer: C, 8x - 5y. Use the identity (A - B)² = A² - 2AB + B². Here 64x² = (8x)² and 25y² = (5y)². The middle term from these parts is -2(8x)(5y) = -80xy, exactly matching the expression. Therefore 64x² - 80xy + 25y² = (8x - 5y)², so the side of the square is 8x - 5y. Option A would produce a positive middle term, +80xy. Option B is not the square root of either visible square term and does not give the required expansion. Option D is twice the correct side, so its square is four times too large. Memory cue: a negative middle term means the two side parts are joined by subtraction; a positive middle term means addition.

The correct answer is B: the two 2x rectangles were omitted. A square of side x+2 can be divided into four parts: an x by x square with area x^2, two rectangles with dimensions x and 2 and area 2x each, and a 2 by 2 corner square with area 4. Therefore, the complete area is x^2+2x+2x+4=x^2+4x+4. The expression x^2+4 includes only the two square parts and misses both rectangular strips. A is wrong because the x^2 part appears once, not twice. C is wrong because the 2 by 2 corner has positive area 4. D is wrong because the given side is x+2, not x-2. Memory cue: in (x+a)^2, the two side strips together always produce 2ax.

The correct answer is A. A square with side x+5 has total area (x+5)^2. In the visual arrangement, this large square consists of an x^2 square, two rectangles of area 5x each, and a 5 by 5 square of area 25. Subtracting x^2+25 removes the two square regions and leaves only the two rectangular strips. Their combined area is 5x+5x=10x, which agrees with (x+5)^2-(x^2+25)=x^2+10x+25-x^2-25=10x. B is wrong because x^2 is specifically removed. C is wrong because 25 is also removed. D is wrong because the two 5x strips remain. Memory cue: subtract the named pieces from the complete picture, not the whole picture.

The correct answer is C. The original square has area a^2, and the removed corner square has area b^2. Thus the remaining area is a^2-b^2. The difference-of-squares identity says a^2-b^2=(a+b)(a-b), so after cutting and rearranging, the pieces form a rectangle whose side lengths are a+b and a-b. A would give area (a-b)^2, which is not generally equal to a^2-b^2. B merely repeats the original dimensions and does not represent the rearranged remainder. D gives area b(a+b), also not the required difference. A useful check is to multiply the proposed sides: (a+b)(a-b)=a^2-b^2. Memory cue: difference of two squares becomes sum times difference.

The correct answer is C. Multiply the two rectangle sides: (x+4)(x+9)=x^2+9x+4x+36=x^2+13x+36. Comparing this with x^2+px+36 shows that p=13. In a visual rectangle, the x by x region gives x^2, the two side strips give 9x and 4x, and the corner gives 4 times 9=36. Thus the coefficient of x is the sum 4+9, not the product. A is the difference 9-4 and does not represent the two strips together. B is the constant product, not the middle coefficient. D is twice the product 4 times 9 and is unrelated here. Memory cue: for (x+a)(x+b), the middle coefficient is a+b and the constant term is ab.

The correct answer is D. Split each side of the square into lengths 3a and 2b. The two equal off-diagonal rectangles each have dimensions 3a by 2b, so each has area (3a)(2b)=6ab. Since there are two such rectangles, their combined area is 6ab+6ab=12ab. This is also the middle term in (3a+2b)^2=(3a)^2+2(3a)(2b)+(2b)^2=9a^2+12ab+4b^2. A is the area of only one rectangle, not both. B is merely the sum of the numerical side coefficients and has no correct area dimensions. C adds the two square areas but leaves out the rectangles. Memory cue: the two middle regions in (u+v)^2 together equal 2uv.

The correct answer is B. Recognize 9m^2=(3m)^2 and 25n^2=(5n)^2. The expression is therefore a difference of squares: (3m)^2-(5n)^2. Applying u^2-v^2=(u+v)(u-v) gives (3m+5n)(3m-5n). A incorrectly uses the full square terms as side lengths instead of their square roots. C would produce (3m-5n)^2, which has a negative middle term and positive 25n^2, but it is not the difference of the two squares. D mixes unsuitable factors and does not multiply back to the original expression. Always verify by multiplication: the cross terms cancel, leaving 9m^2-25n^2. Memory cue: first take the square roots, then write sum and difference.

The correct answer is C. A subtraction square has the form (x-a)^2=x^2-2ax+a^2. Here the total negative middle term is -10x, so 2a=10 and a=5. The two removed strips each have area 5x, consistent with this width. When the two strips are removed, their overlap or missing corner must be restored by a small square of side 5. Its area is 5 times 5=25, so q=25 and the expression is (x-5)^2. A is only the strip width, not its area. B is the total coefficient of the two strips. D incorrectly doubles the required corner area. Memory cue: in x^2-2ax+a^2, the last term is the square of the half-coefficient.

The correct answer is C. The product of two factors represents the area of a rectangle whose side lengths are those two factors. Therefore, (2x+7)(2x-7) is directly shown by a rectangle with sides 2x+7 and 2x-7. Algebraically, it is a difference of squares: (2x+7)(2x-7)=(2x)^2-7^2=4x^2-49, because the two cross-products +14x and -14x cancel. A and B describe squares, whose two sides would be equal; neither represents this product directly. D has area (2x)(7)=14x, not the given product. The visual rectangle may be rearranged into a larger square with corner pieces removed, but the most direct model is the rectangle formed by the two factors. Memory cue: two different factors usually suggest a rectangle; equal factors suggest a square.

The correct answer is A. The large square has side length x+y, so its area is (x+y)^2. Its tiles consist of one x^2 square, two xy rectangles, and one y^2 square. Adding their areas gives x^2+xy+xy+y^2=x^2+2xy+y^2. Therefore, (x+y)^2=x^2+2xy+y^2. B is wrong because the two middle tiles would need negative values for the subtraction identity. C represents a rectangle made from the sum and difference of the sides and has no middle term after cancellation. D is false because the cross-area 2xy is missing from its right side. The two identical xy tiles are the visual reason for the coefficient 2. Memory cue: one first square, two cross rectangles, and one last square means the plus perfect-square identity.

Correct answer: B. The square has side a+b+c, so its area is (a+b+c)². Imagine dividing each side into lengths a, b, and c. The partition contains three smaller squares: a², b², and c². For each pair of different lengths, there are two rectangles: two ab rectangles, two bc rectangles, and two ca rectangles. Therefore the total area is a²+b²+c²+2ab+2bc+2ca. Option A counts only one rectangle for each pair and misses half the area. Option C contains abc, which has three factors and is associated with volume-like multiplication, not a two-dimensional area term. Option D also uses the incorrect abc term and omits the separate pair products. Memory cue: in a three-part square, every pair appears twice; the three square terms appear once.

Correct answer: B. First find the rectangle area: (x+3)(x+5)=x²+5x+3x+15=x²+8x+15. Next find the square area: (x+4)²=x²+8x+16. The requested order is rectangle area minus square area, so subtract: (x²+8x+15)−(x²+8x+16)=−1. Thus the rectangle has area one square unit less than the square. Option A has the correct magnitude but reverses the comparison; it would be square area minus rectangle area. Option C incorrectly leaves an x term after terms that should cancel. Option D is not obtained from area subtraction and resembles an unrelated linear or perimeter expression. Always follow the stated order before deciding the sign.

Correct answer: C. Recognize the expression as a perfect-square trinomial. The constant term 49 is 7², and the middle term is −14x=−2·x·7. Therefore x²−14x+49=(x−7)². In a cut-and-paste area model, the pieces rearrange to form a square whose side is x−7. Option A uses −14 as though it were the side, but −14 is the coefficient of x and must be divided by −2 to identify 7. Option B gives (x+7)², whose middle term is +14x, so its sign is wrong. Option D treats the whole constant 49 as a length, even though 49 is the square of the length 7. Memory cue: check both clues—constant equals the small side squared, and middle coefficient equals twice the signed side.

Correct answer: C. To factor x²+xy−20y², look for two signed numbers whose product is −20 and whose sum is +1. The numbers are +5 and −4. Thus x²+xy−20y²=(x+5y)(x−4y). A visual tile or rectangle model shows x², +5xy, −4xy, and −20y²; the cross terms combine to +xy. Option A gives a product of −20y² but a middle coefficient of 19, not 1. Option B gives the correct constant product, but its cross terms are −5xy+4xy=−xy, not +xy. Option D gives −20y², but its cross terms are −10xy+2xy=−8xy. Signed tiles are important: a negative factor changes the sign of the corresponding rectangular region.

Correct answer: C. Use the square identity (u−v)²=u²−2uv+v² with u=5p and v=2q. The corner associated with the q-part is v²=(2q)²=4q². The two middle rectangles together have area −2(5p)(2q)=−20pq, because the minus sign in the side makes both cross regions negative in the algebraic model. Option A forgets to square 2q, so it gives a length rather than a corner area. Option B halves both required contributions and therefore does not represent the complete pair of middle rectangles. Option D wrongly makes the square of 2q negative; an ordinary square term is positive, while the middle term keeps the negative sign. A useful memory cue is “corner means square, middle means twice the product.”

Correct answer: C. In a rectangle grid, multiply every part of one side by every part of the other side. For (2x+3)(4x+5), the four cells are (2x)(4x)=8x², (2x)(5)=10x, (3)(4x)=12x, and 3·5=15. These are exactly the four given cells, so the sides are possible. Option A produces several unmatched terms and does not give the stated grid. Option B produces 8x², 6x, 20x, and 15; its cross cells are wrong. Option D gives 8x², 4x, 30x, and 15, so its linear cells do not match. The order of the two middle cells is irrelevant because addition is commutative, but their values must be 10x and 12x.

Correct answer: B. Group the first three terms: x²+2xy+y²=(x+y)² by the square identity. The expression therefore becomes (x+y)²−z², which is a difference of two squares. The corresponding model can first show the large square with side x+y and then remove a square of area z². Option A would expand to include terms involving z, such as 2xz and 2yz, which are not present. Option C gives (x−y)²−z²=x²−2xy+y²−z², so the sign of the middle term is wrong. Option D expands to x²+y²−2yz+z², which has different terms and even a positive z². The safe method is to identify the complete square before applying any further identity.

Correct answer: B. In the square identity for three parts, every pair of different side components creates two symmetric rectangles. The pair 2x and z contributes 2(2x)(z)=4xz. The pair 3y and z contributes 2(3y)(z)=6yz. Adding only these requested pairs gives 4xz+6yz. Option A counts one rectangle for each pair and therefore gives only half of each contribution. Option C multiplies all three components and produces xyz, which is not the area of either requested pair. Option D wrongly includes the z² corner square and an unsupported three-factor term; the question asks only for rectangles involving z with 2x and 3y. Memory cue: in a square expansion, a pair of different parts appears twice, while a single part’s corner square appears once.

The direct answer is B: the side is 4x − 5y. Begin by comparing the expression with the identity (a − b)² = a² − 2ab + b². Here, 16x² = (4x)² and 25y² = (5y)². The middle term is −40xy, and −2(4x)(5y) = −40xy, so all three parts match perfectly. Therefore, 16x² − 40xy + 25y² = (4x − 5y)², which is the area of a square whose side is 4x − 5y. A, 4x + 5y, would produce a positive middle term, +40xy. C is not a square root of either first or last term and cannot produce the given trinomial. D has an incorrect coefficient and also does not square to the expression. Memory cue: in a perfect-square trinomial, the sign of the middle term matches the sign between the two parts of the binomial.

The direct answer is C: (3x + 4)(x − 5). A grid model multiplies the side labels, so check every cell: 3x·x = 3x², 3x·(−5) = −15x, 4·x = 4x, and 4·(−5) = −20. Combining the middle cells gives −15x + 4x = −11x, so the product is exactly 3x² − 11x − 20. A gives 3x² + 7x − 20 because −15x + 12x = −3x? More directly, (3x−5)(x+4) = 3x²+7x−20, so its middle term is wrong. B gives 3x² + 11x − 20, with the opposite middle sign. D gives 3x² − 7x − 20, also incorrect. In a signed grid, do not check only the first and last terms; add both cross-products carefully. The exact middle coefficient identifies the correct arrangement.

The direct answer is C: 6xy + 3y². The required area is (x+2y)² − (x−y)². Instead of expanding both squares separately, use a²−b²=(a−b)(a+b). The first factor is (x+2y)−(x−y)=3y, and the second is (x+2y)+(x−y)=2x+y. Multiplying gives 3y(2x+y)=6xy+3y². Thus C is exact. A has only half the required xy coefficient. B has the wrong sign for the y² term, which should be positive because both factors contribute positive y terms. D has an incorrect xy coefficient. A useful check is direct expansion: (x+2y)²=x²+4xy+4y² and (x−y)²=x²−2xy+y²; subtracting gives 6xy+3y². The cancellation of x² is expected in a difference of squares.

The direct answer is D: (3x+5)(4x−1). In a grid, multiply the two main x-parts to get 3x·4x=12x². The cross-products are 3x·(−1)=−3x and 5·4x=20x; together they give 17x. The corner product is 5·(−1)=−5. Thus the complete area is 12x²−3x+20x−5=12x²+17x−5. A gives 12x²+7x−5, so its middle coefficient is wrong. B gives 12x²−17x−5, with the wrong sign for the middle term. C gives 12x²+4x−5, so its cross-products do not total 17x. D matches the leading term, both cross strips, and the constant corner. In a signed grid, always check all three types of terms: leading, middle, and constant.

The direct answer is A. First recognise the first three terms as a perfect square: a²+2ab+b²=(a+b)². The expression therefore becomes (a+b)²−c². Apply U²−V²=(U+V)(U−V), with U=a+b and V=c. The result is (a+b+c)(a+b−c), exactly option A. B is a square of a three-term expression and would introduce extra terms such as −2ac and −2bc. C has the wrong sign pattern and does not reproduce the original first three terms. D expands to ab−ac+bc−c², which lacks a² and b² and therefore cannot be equal. The visual idea is to combine the first three regions into one large square, then remove a c² square and split the difference into two rectangular factors.

The correct answer is A: (x-2y)^2-z^2. First focus on the trinomial x^2-4xy+4y^2. It matches the pattern p^2-2pq+q^2=(p-q)^2, with p=x and q=2y. Therefore it becomes (x-2y)^2. The original expression then is (x-2y)^2-z^2, a difference of two squares. It may subsequently be factored as (x-2y-z)(x-2y+z), though the question asks for the intermediate model. Option A preserves the negative middle term -4xy. Option B would expand to x^2+4xy+4y^2-z^2, with the wrong sign. Option C would require terms involving xz and yz and does not reproduce the original expression. Option D expands to x^2-2xz+z^2-4y^2, which has different variables and terms. Memory cue: a negative middle term in a perfect-square trinomial gives (first term minus second term)^2.