Visual models of identities

The correct answer is D: 4m² − 25. Begin with the basic fact that the area of a rectangle is length multiplied by breadth. Therefore, the area is (2m + 5)(2m − 5). These two factors have the special form (a + b)(a − b), called conjugate factors. The identity is (a + b)(a − b) = a² − b². Here a = 2m and b = 5, so the area is (2m)² − 5² = 4m² − 25. We can also verify it by distribution: 2m·2m = 4m², 2m·(−5) = −10m, 5·2m = 10m, and 5·(−5) = −25. The middle terms −10m and +10m cancel, leaving 4m² − 25. A is wrong because the constant term must be negative. B is wrong because (2m)² is 4m², not 2m². C is wrong because it does not correctly cancel the middle terms and also has the wrong constant sign. D is correct. Memory cue: conjugate factors produce a difference of squares, not a sum of squares.

The correct answer is B: 28. Use the side lengths x + 4 and x + 7. Splitting each side gives four products: x·x = x², x·7 = 7x, 4·x = 4x, and 4·7 = 28. The small corner is formed by the two constant pieces, 4 and 7, so its area is 4·7 = 28. Option A, 11x, is the combined area of the two mixed rectangles because 7x + 4x = 11x. Option C is only one mixed rectangle. Option D is the other mixed rectangle. Therefore the full expansion is (x + 4)(x + 7) = x² + 11x + 28. A useful check is that a corner made from two constants has no x in its area.

The correct answer is C: 12x. Split the side 2x + 3 into the parts 2x and 3. In the square area model, the two mixed rectangles each have dimensions 2x and 3, so each area is (2x)(3) = 6x. Since there are two such rectangles, their combined area is 6x + 6x = 12x. This is the middle term in (2x + 3)²: using 2uv with u = 2x and v = 3 gives 2(2x)(3) = 12x. Option A gives only one mixed rectangle. Option B does not use the variable part 2x correctly. Option D is the area of the variable square (2x)² = 4x², not the two mixed rectangles. Remember: “two mixed rectangles” means twice the product of the two side parts.

The direct answer is A: \((a-b)^2=a^2-2ab+b^2\). Begin with the original square, whose area is \(a^2\). Each removed strip has dimensions \(a\times b\), so each has area \(ab\). Removing both strips gives \(a^2-2ab\). However, the two strips overlap in the corner square of area \(b^2\); that corner was subtracted twice, so it must be added once. The remaining figure is a square whose side is \(a-b\), and its area is therefore \((a-b)^2\). Option B is the difference-of-squares identity and involves two complete squares, not this two-strip removal. Option C is the square-of-a-sum identity, so it has a positive middle term. Option D has no valid general area interpretation and fails by expansion. Memory cue: remove two rectangles, then add their overlap once.

The direct answer is B: \(9x^2-24x+16\). In a square model, the corner square areas \(9x^2\) and \(16\) have side lengths \(3x\) and \(4\), because \((3x)^2=9x^2\) and \(4^2=16\). A negative middle term means the outer side is \(3x-4\), not \(3x+4\). Squaring gives \((3x-4)^2=(3x)^2-2(3x)(4)+4^2=9x^2-24x+16\). Option A uses the correct magnitudes but the wrong positive sign. Option C incorrectly squares or changes the first side and has a middle coefficient of \(-8\), not \(-24\). Option D omits the middle area and is a difference of squares, not a square. Memory cue: the sign of the middle term tells whether the two side lengths are added or subtracted.

The direct answer is B: the possible sides are \((x+4)\) and \((x+5)\). A rectangle model for a quadratic expression represents multiplication of two binomials. The constant corner must come from multiplying the constant parts, so we need two numbers whose product is 20. The coefficient of \(x\) comes from their sum, so those same numbers must add to 9. The pair 4 and 5 satisfies both tests: \(4\times5=20\) and \(4+5=9\). Therefore \((x+4)(x+5)=x^2+9x+20\). Option A has product 20 but sum 12, giving \(x^2+12x+20\). Option C has product 20 but sum 21. Option D has sum 9 but product 18. Memory cue: for \(x^2+sx+p\), check factor product \(p\) and factor sum \(s\).

The direct answer is C: \(-10pq\). The square has side \(5p-q"). Treat its two parts as \(u=5p\) and \(v=-q\). The square identity is \((u+v)^2=u^2+2uv+v^2\). The two middle rectangles together have area \(2uv\), so here it is \(2(5p)(-q)=-10pq\). The negative sign appears because one part of the side is a subtraction. Option A forgets the factor 2 and also misses the negative sign. Option B includes the factor 2 but treats \(-q\) as positive. Option D is the square of \(5p\), namely the first corner square, not the combined middle area. The full expansion is \(25p^2-10pq+q^2\). Memory cue: two middle rectangles give twice the product, and a minus side makes that product negative.

The direct answer is B: \(4a+5b\). Recognize the expression as a perfect-square trinomial. The first term is \(16a^2=(4a)^2\), and the last term is \(25b^2=(5b)^2\). If the outer side is \(4a+5b\), the middle term should be \(2(4a)(5b)=40ab\), which is exactly the given middle term. Therefore \(16a^2+40ab+25b^2=(4a+5b)^2\), so the side is \(4a+5b\). Option A adds the two corner areas rather than taking their side lengths. Option C would square to a first term of \(64a^2\), so it is too large. Option D would produce a middle term of \(80ab\), not \(40ab\). Memory cue: take square roots of the first and last terms, then check twice their product.

The direct answer is C: \(17x\). Use the rectangle model to multiply each part of the first factor by each part of the second factor: \((3x)(x)=3x^2\), \((3x)(5)=15x\), \(2(x)=2x\), and \(2(5)=10\). The two parts containing \(x\) are \(15x\) and \(2x\). They are like terms, so their coefficients are added: \(15x+2x=17x\). Thus the complete product is \(3x^2+17x+10\). Option A gives only one of the two linear regions. Option B gives only the other linear region. Option D is the constant product \(2\times5"), not an \(x\)-term. The area model prevents missing a product because it displays all four combinations. Memory cue: in a two-binomial product, collect the two cross-products to obtain the middle term.

The direct answer is A: \(4x^2-12x+9\). Apply the square-of-a-difference identity \((u-v)^2=u^2-2uv+v^2\), with \(u=2x\) and \(v=3\). The first square is \((2x)^2=4x^2\), the two middle rectangles together have area \(-2(2x)(3)=-12x\), and the small corner square has area \(3^2=9\). Adding these signed parts gives \(4x^2-12x+9\). Option B uses \(-6x\) and misses the factor 2. Option C incorrectly changes \((2x)^2\) to \(2x^2\); the coefficient must also be squared. Option D has the wrong positive sign for the middle term and represents \((2x+3)^2\). Memory cue: square the coefficient, double the product, and keep the minus sign for the cross-term.

The direct answer is B: \(a+b=16\). The difference-of-squares identity is \(a^2-b^2=(a-b)(a+b)\). In a cut-and-paste model, the region between the squares can be rearranged into a rectangle whose sides are \(a-b\) and \(a+b\), so its area is their product. Substitute the given values: \(96=6(a+b)\). Divide both sides by 6 to obtain \(a+b=96\div6=16\). Option A, 90, incorrectly subtracts 6 from 96; area is not used that way here. Option C, 102, incorrectly adds 6 to 96. Option D, 12, fails because \(6\times12=72\), not 96. The key is to use the factorisation, not ordinary addition or subtraction of the given numbers. Memory cue: difference of squares equals difference times sum.

The direct answer is A: two rectangles, each having area ab. Imagine a large square whose side is split into lengths a and b. The large square has four regions: one square of area a², one square of area b², and two rectangular regions. Each rectangle has one side a and the other side b, so each has area ab. Adding all four areas gives a²+ab+ab+b² = a²+2ab+b². The student included the two square regions but omitted both middle rectangles, which together contribute 2ab. Option A is correct because it identifies exactly the missing regions. Option B is wrong because the a² square is already included. Option C is wrong because the b² square is also already included. Option D is wrong because the outer side is a length used to form the model, not an omitted area region. Memory cue: the middle term in (a+b)² comes from two cross-rectangles.

The direct answer is C: 16x²−9y². Put A=4x and B=3y. The expression then becomes (A+B)(A−B), which is the difference-of-squares identity: (A+B)(A−B)=A²−B². Substituting back gives (4x)²−(3y)² = 16x²−9y². Expansion confirms the result: 4x·4x=16x², 4x·(−3y)=−12xy, 3y·4x=12xy, and 3y·(−3y)=−9y². The two middle terms cancel. A is wrong because it adds the squares, which is not the identity here. B has incorrect coefficients and powers. D incorrectly leaves a middle term −24xy; that would arise from multiplying expressions with the same negative sign pattern, not opposite signs. Memory cue: same first part and opposite second parts means square the parts and subtract.

The direct answer is B: 121. In the visual model for (x+a)², each of the two rectangles has side lengths x and a, so its area is ax. We are told that ax=11x. For x not equal to zero, comparing the common factor x gives a=11. The remaining small square has both sides equal to a, so its area is a²=11²=121. A is only 2a, not the area of the small square. C incorrectly keeps x²; x² is the area of the large x-square, not the small constant square. D is 2ax, which represents the combined area of the two rectangles, not the final square. The model therefore contains x², 11x, 11x and 121. Memory cue: one rectangle gives ax; the last small square gives a².

The direct answer is B: 12yz. Divide each side of the large square into lengths x, 2y and 3z. A rectangle containing the product yz must use the 2y strip in one direction and the 3z strip in the other direction. Its area is (2y)(3z)=6yz. Because the square is symmetric, there are two such rectangles: one above or beside the corresponding region and its matching partner across the diagonal. Their combined area is 6yz+6yz=12yz. A, 5yz, incorrectly adds the coefficients 2 and 3 instead of multiplying them. C is the area of only one yz rectangle, not both. D ignores the factor 2 and does not represent either the single or combined area correctly. Memory cue: in a square expansion, every pair of different side parts produces two matching rectangles.

The direct answer is C: k=7. The square model represents (x+k)². Using the identity (x+k)²=x²+2kx+k², compare it with x²+14x+49. The constant term tells us k²=49, so the positive side length gives k=7. The middle term checks the result: 2kx=2·7·x=14x, exactly as required. A=14 is the coefficient of x, not k; it would make the middle term 28x. B=21 has no matching square or middle coefficient. D=49 is the constant term, but k is its square root, not the term itself. Because a side length is taken as positive in this model, 7 is the suitable value. Memory cue: for x²+mx+n, first take k=√n and then verify that 2k=m.

The direct answer is B: 4ab. The area of the larger square is (a+b)²=a²+2ab+b². The area of the other square is (a−b)²=a²−2ab+b². Subtracting the second from the first gives (a²+2ab+b²)−(a²−2ab+b²). The a² terms cancel, the b² terms cancel, and the two middle contributions add: 2ab−(−2ab)=4ab. Thus the remaining visual area is 4ab. A, 2ab, is only the middle term from one square, not the difference of both areas. C is the sum of the square terms and does not result after cancellation. D is the difference-of-squares factorisation result for (a+b)(a−b), a different expression. Memory cue: subtracting the two square expansions doubles the opposite middle terms, giving four ab.

The direct answer is option B: 16x. A square model with corner areas x² and 64 has side lengths x and 8, because 64 = 8². When these two lengths form the sides of a larger square, the two middle rectangular regions each have area x × 8 = 8x. The combined middle area is therefore 8x + 8x = 16x. Algebraically, this is the identity (x + 8)² = x² + 16x + 64. Option A gives only one of the two rectangles, not their combined area. Option C doubles 16x again and is too large. Option D treats 64 as the middle term instead of as the corner square. Remember: the middle coefficient in (p + q)² is 2pq; first take the square root of 64 to obtain 8, then calculate 2 × x × 8.

The direct answer is option A. Divide the rectangle into four parts by using x and y along one direction and x and −y along the other. The four signed areas are x·x = x², x·(−y) = −xy, y·x = xy, and y·(−y) = −y². The two middle contributions are equal in magnitude but opposite in sign: xy + (−xy) = 0. Therefore the xy term disappears, leaving x² − y². Option B is wrong because two xy-shaped regions are formed; their values cancel rather than fail to exist. Option C is wrong because x² does not become zero. Option D is wrong because the y² contribution is negative here, not the reason for cancellation. The identity is (x+y)(x−y)=x²−y². A useful warning is that unlike terms cannot cancel; only like terms with opposite signs, such as xy and −xy, can do so.

The direct answer is option B: negative 4x. Expand the two factors by pairing every term in the first factor with every term in the second: x·x = x², x·2 = 2x, (−6)·x = −6x, and (−6)·2 = −12. The terms containing exactly one x are 2x and −6x. They are like terms, so combine their coefficients: 2x − 6x = −4x. Thus the combined x-term is −4x. Option A, 8x, adds the magnitudes and loses the signs. Option C, +4x, has the correct magnitude but the wrong sign. Option D, −8x, incorrectly doubles the magnitude. The constant −12 is a separate term and must not be included in the requested x-term. In a signed area model, the negative rectangle contributes a negative area, so its sign remains visible. Memory cue: cross-products make the middle terms; keep each factor’s sign before combining.

The direct answer is option C: x−9. The expression x² − 18x + 81 is a perfect square. Since 81 = 9² and the middle coefficient −18 equals −2·x·9, factor the expression as x² − 2·9x + 9² = (x−9)². In a square model, the expression for the total area is the square of its side, so the side is x−9. Option A, x+9, would produce x² + 18x + 81, with the wrong sign in the middle. Option B, x−18, would not square to the given expression; it uses the middle coefficient as a side number. Option D, x+18, gives both the wrong sign and the wrong constant term. The last term helps identify 9, while the middle term tells whether the sign is plus or minus. A useful check is to multiply (x−9)(x−9): the cross-products are −9x and −9x, which combine to −18x.

The direct answer is option A. Begin with the large square of area a². Removing one ab rectangle subtracts ab, and removing two subtracts 2ab, giving a² − 2ab. The two removed regions overlap in a b-by-b square, whose area is b². That common part has been subtracted twice, so it must be added back once. The final area is a² − 2ab + b², which is exactly (a−b)². Option B has positive cross-terms and represents adding strips rather than removing them. Option C is the difference-of-squares identity; it contains no 2ab term and describes a product of two different factors. Option D has an incorrect expansion: (a+b)(a−b) equals a²−b², not the expression shown. The inclusion–exclusion idea is important in the visual model: subtract both removed rectangles, then restore their overlap once. Memory cue: a square of a difference has two negative cross-terms and one positive small square.

The direct answer is option A. Expand both square expressions or subtract them carefully: (x+y)² = x² + 2xy + y², while (x−y)² = x² − 2xy + y². Subtracting the second from the first gives [(x+y)² − (x−y)²] = 4xy. Rearranging yields (x+y)² = (x−y)² + 4xy. Thus the larger square-of-a-sum area consists of the square-of-a-difference area plus the additional 4xy area. Option B reverses the two square areas and would require 4xy to be negative in the usual positive-area setting. Option C adds the two squares; their sum is 2x² + 2y², not generally 4xy. Option D uses the wrong sign. A quick memory cue is: the difference between the square of a sum and the square of a difference is four times the product, so the plus-square equals the minus-square plus 4xy.

The direct answer is A: a² + 2ab + b² = (a + b)². Begin with the original square, whose area is a². The strip on the right has dimensions a by b, so its area is ab. The strip at the bottom also has area ab because it has the same dimensions. The little corner is a b by b square, so its area is b². Adding all non-overlapping parts gives a² + ab + ab + b² = a² + 2ab + b². These pieces together form a larger square with side a+b, whose area is (a+b)². B describes a difference of squares, not an enlarged square. C is used when widths are removed, so its middle term is negative. D counts only one ab strip and therefore misses part of the area. Memory cue: adding the same width to both sides gives plus 2ab.

The direct answer is A: p² − 2pq + q². The original square has area p². Each removed strip has one dimension p and the other q, so each has area pq. Removing both strips gives p² − pq − pq, or p² − 2pq. However, the top-right corner belongs to both strips. It was therefore subtracted twice even though it should be removed only once. Add that overlapping q by q square back once: p² − 2pq + q². This expression is also (p−q)², which makes sense because the remaining central square has side p−q. B has positive strip terms and represents addition rather than removal. C subtracts the overlap correction incorrectly and loses the two strip areas. D has no correct area decomposition. Memory cue: in overlapping subtraction, subtract both parts and add their common overlap once.