Visual models of identities

Direct answer: Option A. The outside figure is a square whose side is x+y, so its total area is (x+y)². Inside it are an x by x square with area x², a y by y square with area y², and two x by y rectangles. Together the rectangles have area xy+xy=2xy. Adding all pieces gives x²+2xy+y², which must equal the area of the whole square. Thus (x+y)²=x²+2xy+y². B describes a subtraction arrangement and has a negative middle term. C and D are the difference-of-squares identity, not a square containing two positive xy rectangles. Common warning: two equal positive rectangles mean +2xy, not −2xy.

Direct answer: A, x+6. The parts of the model give the total area x²+12x+36. For a square with side x+r, the area is (x+r)²=x²+2rx+r². Here the constant part 36 is 6², so r=6. The middle part is then 2·x·6=12x, exactly as shown. Therefore x²+12x+36=(x+6)², and the side length is x+6. A is correct because its square reproduces all three parts. B would give (x+12)²=x²+24x+144, so both the middle and constant terms are wrong. C would give (x+36)²=x²+72x+1296, which is also incorrect. D would give (x−6)²=x²−12x+36; although its first and last terms match, its middle term has the wrong sign. Memory cue: halve the coefficient of x and check that its square equals the constant term.

Direct answer: Option B, 10. In the rectangle model, split one side into p and 2 and the other side into p and 5. This creates four smaller rectangles. Their areas are p·p=p², p·5=5p, 2·p=2p, and the corner region 2·5=10. The corner rectangle uses the two constant side lengths, 2 and 5, so its area is 10. Option A is the area of the rectangle with sides 2 and p. Option C is the area of the rectangle with sides p and 5. Option D is not a product of the two corner lengths and does not correspond to any region in this model. Memory cue: the corner formed by the constant pieces is found by multiplying constants, 2×5.

Direct answer: C, a² + 4ab + 4b². The area of a square is side × side, so the required area is (a+2b)². Use the identity (p+q)²=p²+2pq+q² with p=a and q=2b. Thus, (a+2b)²=a²+2(a)(2b)+(2b)²=a²+4ab+4b². In an area model, the large corner square contributes a², the two equal rectangles each contribute 2ab, and the small corner square contributes 4b². A is wrong because its middle term is only 2ab. B leaves out the two rectangular regions. D has both an incorrect middle term and an incorrect final term. Memory cue: square the first term, double the product, and square the second term.

Direct answer: D, (c+d)(c-d). Begin with a large square of side c, whose area is c². Remove a smaller square of side d, whose area is d². The remaining region has area c²-d². If this remaining region is rearranged into a rectangle, its two dimensions are c+d and c-d. Therefore, c²-d²=(c+d)(c-d), the difference-of-squares identity. A gives only c(c-d), which does not include the complete factor c+d. B has the wrong factors and represents a different product. C uses addition rather than subtraction, so it is the sum of two squares, not the remaining area. D is correct because multiplying it gives c²-cd+cd-d²=c²-d². Cue: difference of squares means sum of terms times their difference.

Direct answer: C, (x+y)²=x²+2xy+y². The full figure is a square because both its total length and total breadth are x+y. Its area is therefore (x+y)². Dividing it into four regions gives one x-by-x square with area x², two x-by-y rectangles with areas xy and xy, and one y-by-y square with area y². Adding these areas gives x²+xy+xy+y²=x²+2xy+y². A is the identity for subtracting lengths, so its middle term is negative. B describes a difference of squares and has no two positive xy rectangles. D has incorrect signs and does not describe these four positive regions. Memory cue: two added rectangular parts produce +2xy.

Direct answer: B, 9. A square whose side is t−3 has area (t−3)². Use the identity (p−q)²=p²−2pq+q². Substituting p=t and q=3 gives t²−2(t)(3)+3²=t²−6t+9. The last term is the constant term q², so it is 9. In an area model, this term represents the small square with side 3; its area is 3×3=9, which cannot be negative. A, −9, is wrong because squaring 3 gives a positive result; the minus sign affects the two rectangular middle parts. C, 3t, contains a variable and represents neither the final constant term nor the complete middle contribution. D, −6t, is the middle term formed by combining two products of t and −3. Thus B is the only valid answer. Memory cue: in (a−b)², the final term is always b² and is positive.

Direct answer: C, (u+2)(u+5). In a rectangular area model, the four smaller areas come from multiplying each part of one side by each part of the other side. The u² region suggests u×u. The 2u region is u×2, and the 5u region is 5×u. The remaining corner is 2×5=10. Therefore the side lengths are u+2 and u+5, so the whole product is (u+2)(u+5). Expanding confirms it: u²+5u+2u+10=u²+7u+10. A has unmatched constants and does not produce the listed four regions. B would have two 2u regions and a corner 4. D would have two 5u regions and a corner 25. The corner product is a useful checking method.

Direct answer: D, 2x+3. The expression is the area of a square, so we need its side whose square is 4x²+12x+9. The first term 4x² is (2x)², and the last term 9 is 3². If the side is 2x+3, the cross-term is 2(2x)(3)=12x, exactly the middle term. Hence 4x²+12x+9=(2x+3)². A, 2x+9, would produce (2x+9)²=4x²+36x+81, not the given expression. B, 4x+3, would begin with (4x)²=16x², so its leading term is wrong. C, 2x+6, would produce a middle term 24x and constant term 36. D matches the first term, middle term, and last term simultaneously. The safest method is to check all three parts, especially 2ab, rather than using only the first and last terms.

Direct answer: A, a square of side a. In the area model for (a+b)², each full side is split into lengths a and b. The region where the a part meets the a part has dimensions a by a. Its area is a×a=a², so it must be a square whose side is a. B describes the b² corner region. C has area ab, so it represents one of the two rectangular cross-regions, not a². D refers to the outside figure and does not describe the a² subregion. The complete model contains one a² square, two ab rectangles, and one b² square, giving a²+2ab+b². A common confusion is to read a² as a side length; it is an area, and the corresponding side is a.

Direct answer: B, 3x. Split each side of the square into x and 2. The four regions are formed by pairing these lengths: x×x=x², x×2=2x, 2×x=2x, and 2×2=4. Thus the visual model contains one x² square, two rectangles each with area 2x, and one small square with area 4. The expansion confirms this: (x+2)²=x²+2x+2x+4=x²+4x+4. A appears as the large x-by-x square. C appears twice as the two equal rectangles. D appears as the 2-by-2 corner square. B, 3x, is not the area of any individual region. Even though terms can sometimes be combined algebraically, the model’s separate regions are determined by the actual side partitions, so 3x does not appear. Do not confuse the combined 4x with an individual 3x region.

Direct answer: C, difference of squares. A square of side a has area a². A smaller square of side b has area b². When the smaller square is removed, subtraction is required, so the remaining area is a²-b². This expression is called the difference of squares. A, square of a sum, refers to (a+b)² and represents a different construction in which lengths are added before squaring. B, a perfect-square trinomial, usually has three terms such as a²-2ab+b² and represents (a-b)², not simply one square area removed from another. D means a²+b², which adds the two areas rather than removing one. The visual action is the key: removed area means subtraction. If needed, the remaining region can also be rearranged to show a²-b²=(a+b)(a-b).

Direct answer: D, x²+6x. The area of a rectangle is length multiplied by breadth. Here the area is (x+6)×x. Use the distributive property: x(x+6)=x·x+x·6=x²+6x. In the area model, the x-by-x region contributes x² and the 6-by-x region contributes 6x; adding them gives x²+6x. A is wrong because the 6 must also be multiplied by x. B multiplies incorrectly and gives an x² term with coefficient 6. C is not the correct distributive expansion and does not represent the product of the two dimensions. D matches both the geometry and the algebra. A helpful check is to substitute x=1: the rectangle has dimensions 7 and 1, so its area is 7; D gives 1+6=7.

Direct answer: A, 2 rectangles. Divide each side of the large square into lengths s and 4. The division creates four parts: one square of area s², two rectangles with dimensions s by 4, and one square of area 16. Each of the two rectangles has area s × 4 = 4s. Therefore, exactly two regions have area 4s. Option A is correct. Option B is wrong because there are two strips, one along each direction. Option C confuses the number 4, which is a length, with the number of rectangles. Option D is too large and does not match the four-part area model. Algebra confirms this: (s+4)² = s² + 4s + 4s + 16 = s² + 8s + 16. Memory cue: in (a+b)², the two middle regions are ab and ab.

Direct answer: B, the square-of-difference model. The outer square has side a, so its area is a². If strips of width b are removed from the appropriate sides, the side left for the inner square is a − b. Its area is therefore (a − b)². This is exactly the visual meaning of the identity (a − b)² = a² − 2ab + b². Option A is incorrect because the square-of-sum model uses a side a+b and produces (a+b)². Option C describes multiplying the dimensions of a rectangle, but it does not identify this special remaining-square arrangement. Option D concerns adding separate square areas and does not show an inner square obtained by subtraction. Memory cue: a reduced side a−b signals “difference,” while an enlarged side a+b signals “sum.”

Direct answer: A, x² + y² + 2xy. Add the area of every tile. The x by x square contributes x², the y by y square contributes y², and each x by y rectangle contributes xy. Since there are two such rectangles, together they contribute 2xy. The total area is x² + 2xy + y², which is the identity (x+y)². Option B counts only one xy rectangle. Option C uses subtraction and would not match a square whose side is x+y. Option D counts two y² squares and omits both the correct middle contribution and the x-by-y structure. A useful check is to expand (x+y)²: the middle term appears twice because the two cross-rectangles are congruent.

Direct answer: D, (a+b)². The area of a square is side multiplied by itself. Since the outer boundary has side length a+b, its total area is (a+b)(a+b) = (a+b)². This is the direct expression obtained from the outside of the model. Option A is also equal to the same area after the square is divided into parts and the identity is expanded, but it is not the form read directly from the outer boundary. Option B represents a difference of squares, not this complete square. Option C is only the product of a and b and ignores most of the model. Memory cue: outside side gives the compact square form; inside pieces give the expanded form a²+2ab+b².

Direct answer: A, because both expressions represent the area of the same square. In the model, the outer square has side a+b, so its area is (a+b)². If the square is divided into one a-by-a square, two a-by-b rectangles, and one b-by-b square, the sum of the parts is a²+ab+ab+b² = a²+2ab+b². The pieces exactly fill the same outer square, so their total area must equal the area calculated from its full side. B is wrong because the expressions concern area, not perimeter. C is wrong because the expressions are not always zero. D is wrong because they describe areas, not merely lines. The diagram provides a visual proof of the identity (a+b)² = a²+2ab+b².

Direct answer: B, x²−4. The factors x+2 and x−2 are conjugates: they have the same first term and opposite second terms. Apply (a+b)(a−b)=a²−b² with a=x and b=2. This gives x²−2²=x²−4. Direct multiplication confirms it: x·x+x(−2)+2x+2(−2)=x²−2x+2x−4=x²−4. The middle terms cancel because −2x and +2x are opposites. A, x²+4, has the wrong sign for the constant product 2(−2), which is −4. C, x²−2x, keeps only one middle product and omits the constant term. D, x²+2x, also leaves an unjustified middle term and omits −4. Therefore B is the only complete simplification. Memory cue: multiplying conjugate binomials gives a difference of squares and no middle term.

Direct answer: C, from the square of b. A square whose side length is b has area b×b=b². In the larger square with side 2a+b, the b portion on one direction combines with the b portion on the other direction to create the small b-by-b square. Option A gives (2a)b=2ab, which is a rectangular cross-term, not b². Option B gives (2a)²=4a², the square of the other portion. Option D is a sum of lengths, not an area, and it does not produce the term b². A useful rule is that a squared term such as b² comes from a square tile whose two dimensions are both b.

Direct answer: A, areas ac and bc. Split the side a+b into two lengths, a and b, while keeping the other side c unchanged. This divides the original rectangle into an a-by-c rectangle with area ac and a b-by-c rectangle with area bc. Adding them gives the full area c(a+b)=ac+bc, which is the distributive property. B is wrong because ab uses the two pieces of one side and c² does not belong to either resulting rectangle. C would describe two squares, but the model contains rectangles with common side c. D lists side lengths, not areas; area requires multiplication of two lengths. Memory cue: when a sum is multiplied by c, distribute c to each term: c(a+b)=ca+cb.

Direct answer: B, x−3. Compare the expression with the identity (a−b)²=a²−2ab+b². Taking a=x and b=3 gives (x−3)²=x²−2(x)(3)+3²=x²−6x+9. Thus the displayed area belongs to a square with side x−3. Multiplication verifies the result: (x−3)(x−3)=x²−3x−3x+9=x²−6x+9. A, x+3, would give x²+6x+9, so its middle term has the wrong sign. C, x−6, would give x²−12x+36, with both the middle coefficient and constant term wrong. D, x+9, would give x²+18x+81, which is also different. The constant 9 identifies 3², while the negative middle term tells us that 3 is subtracted rather than added. Memory cue: for a perfect square, the sign of the middle term determines the sign inside the binomial.

Direct answer: A. The whole square has side p+q, so its area is (p+q)². The partition contains one p-by-p square with area p², one q-by-q square with area q², and two p-by-q rectangles. The two rectangles contribute pq+pq=2pq. Adding all regions gives p²+2pq+q², so (p+q)²=p²+2pq+q². B is the square-of-difference identity; it would have negative cross-terms and a side p−q. C is the difference-of-squares identity and involves a product of two different binomials, not this four-region sum. D is false in general because the cross-terms do not disappear in this model. Memory cue: plus side means plus twice the product in the middle.

The direct answer is C, 84 square units. The area of a square is found by multiplying its side by itself, so the large square has area 10² = 100 square units. The small square has area 4² = 16 square units. The difference of their areas is therefore 100 − 16 = 84 square units. This is the identity a² − b² when a = 10 and b = 4. Option A, 40, is only 10 × 4 and does not represent the difference of the two square areas. Option B, 60, comes from 10² − 4, where the small side was not squared. Option C is correct because it subtracts both complete areas. Option D, 116, adds 100 and 16 instead of finding the difference. Remember: square both side lengths before subtracting.

The direct answer is D, 64. A square whose side is x+8 has area (x+8)². Use the identity (a+b)²=a²+2ab+b², with a=x and b=8. The expansion is x²+2(x)(8)+8²=x²+16x+64. The last term is therefore 64, because it comes from squaring the constant part 8. Option A, 8x, is only a product involving x and is not the final constant term. Option B, 16x, is the middle term produced by the two equal cross-products x·8 and 8·x. Option C, x², is the first term, obtained by squaring x. Option D is correct because 8²=64. In an area model, the two side regions give the middle contribution and the small constant square gives the last term. Memory cue: in (variable+number)², the final term is the number squared.