Visual models of identities

The direct answer is A: two rectangles, each having area xy. Imagine a square whose horizontal and vertical side lengths are both split into parts x and y. These two splits create four smaller regions. The corner where the x parts meet is an x by x square with area x^2. The corner where the y parts meet is a y by y square with area y^2. The two remaining regions each have one side x and the other side y, so each is an xy rectangle. Adding all four regions gives (x+y)^2=x^2+xy+xy+y^2=x^2+2xy+y^2. A is therefore correct. B is a line, not an area, so it cannot be one of the rectangular pieces. C incorrectly duplicates the x^2 square and ignores the two cross-regions. D uses x-y, which is not a side length created by this partition. Memory cue: in the plus-square model, the middle term 2xy comes from two equal cross-rectangles.

The direct answer is B: a square with side x+4. A square is a quadrilateral with four equal sides and right angles. In the given rectangle, the length is x+4 and the breadth is also x+4. Since the two adjacent dimensions are equal, the opposite dimensions are equal as well, so all four sides have length x+4; the rectangle is therefore a square. Its area is (x+4)(x+4)=(x+4)^2, and expanding gives x^2+8x+16. A describes a generally different rectangle with area 4x, not the given area. B repeats both given dimensions and is correct. C has a different side length and would have area (x-4)^2. D has unequal dimensions x+8 and x, so it is not the same figure or area in general. The useful rule is: when a rectangle’s length equals its breadth, it is a square; the common dimension becomes the square’s side.

The direct answer is C: 2ab. Start with a large square of side a, whose area is a^2. To reduce each side from a to a-b, remove one strip of width b and length a along one direction, and another strip of width b and length a along the other direction. Each strip has area ab, so the total rectangular area removed is ab+ab=2ab. The small b by b corner is removed twice in that description, so it must be added back once; this gives a^2-2ab+b^2=(a-b)^2. A counts only one strip and misses the second. B is the small corner, not the total rectangular subtraction; it is added back. C correctly counts both strips. D is the entire original square, not the removed part. Memory cue: the negative middle term -2ab represents two removed cross-strips, while +b^2 repairs the overlap.

The direct answer is A, 100. The area of a square is side multiplied by itself, so the required area is (7+3)^2. Use the identity (a+b)^2=a^2+2ab+b^2 with a=7 and b=3. Then (7+3)^2=7^2+2(7)(3)+3^2=49+42+9=100. A is correct. B, 49, is only 7^2 and leaves out the contribution of 3 and the two cross-products. C, 79, does not result from the identity and is numerically incorrect. D, 91, also omits or miscalculates part of the middle and final terms. A quick check is possible without the identity: 7+3=10, and 10^2=100. The visual meaning is that the large square contains one 7-by-7 square, two 7-by-3 rectangles, and one 3-by-3 square. Memory cue: for a squared sum, never forget the middle term 2ab; all three pieces a^2, 2ab, and b^2 are needed.

The direct answer is B. Compare the expression with the identity a^2-2ab+b^2=(a-b)^2. Here a=12 and b=2, so 12^2-2×12×2+2^2=(12-2)^2. A square whose side is 12-2 therefore has exactly this area. A would correspond to (12+2)^2, whose middle term is positive, not negative. C would produce a square with side 24 and area 576, which is unrelated to the given expression. D would mean a side of 144 and an extremely different area; it confuses a number with its square. Option B is also easy to check numerically: the expression equals 144-48+4=100, and (12-2)^2=10^2=100. The visual subtraction model starts with the 12-by-12 square, removes two strips of width 2, and restores the 2-by-2 overlap. Memory cue: a negative middle term signals (a-b)^2; a positive middle term signals (a+b)^2.

The direct answer is A. Use the difference-of-squares identity a^2-b^2=(a+b)(a-b). Substituting a=9 and b=4 gives 9^2-4^2=(9+4)(9-4)=13×5. In a visual model, a 9-by-9 square and a 4-by-4 square are compared; the remaining pieces can be rearranged to form one rectangle with side lengths 9+4 and 9-4. A therefore describes the resulting rectangle. B gives 9×4, which is a product of the original side lengths, not the difference of their square areas. C gives dimensions 18 and 8, whose product is 144 and does not equal 65. D gives a 5-by-5 square, not the rearranged rectangle, and its area is 25. A numerical check confirms the result: 81-16=65 and 13×5=65. Memory cue: difference of two squares becomes a sum times a difference—plus first, minus second.

The direct answer is B: x^2+6x. Think of x(x+6) as the area of a rectangle with one side x and the other side x+6. Split the second side into lengths x and 6. The original rectangle is then divided into two smaller rectangles. The first has dimensions x by x, so its area is x^2. The second has dimensions x by 6, so its area is 6x. Adding the two parts gives x^2+6x, which is the distributive property x(x+6)=x·x+x·6. A, x^2+36, incorrectly treats 6 as if it were multiplied by itself; the second part should be 6x. C changes the terms and gives the wrong degree and coefficients. D is only the total side length x+6, not an area, because it lacks multiplication by the other side x. B is confirmed by a quick expansion and by the rectangle picture. Memory cue: split a side, then multiply the unchanged side by each piece; areas add.

The direct answer is C: two. Divide the side a+b into segments a and b in both directions. This partitions the large square into four regions: one a-by-a square with area a^2, two cross-rectangles with dimensions a and b and area ab each, and one b-by-b square with area b^2. Therefore the full area is (a+b)^2=a^2+ab+ab+b^2=a^2+2ab+b^2. The coefficient 2 in 2ab is not accidental; it counts the two rectangular ab parts. A counts only one cross-rectangle and misses the other. B counts three, but the third region is a square, not an ab rectangle. C is correct because exactly two regions have side lengths a and b. D incorrectly treats all four regions as ab rectangles, although two are squares. Memory cue: the middle term 2ab means two identical cross-rectangles in the plus-square diagram.

The direct answer is D: x+y. Recognize the expression as a perfect-square identity: x^2+2xy+y^2=(x+y)^2. In a tile model, the x^2 tile is a square with side x, the y^2 tile is a square with side y, and the two xy tiles are rectangles with sides x and y. These four tiles fit together to form one larger square. Since its area is (x+y)^2, the side of that square is x+y. A is incorrect because (x-y)^2=x^2-2xy+y^2, which has a negative middle term. B is the area of one xy rectangle, not the outer side. C would produce (2x+y)^2=4x^2+4xy+y^2, which is not the given expression. D matches both the tile arrangement and the identity. A reliable method is to take the square roots of the first and last terms, x and y, and check that the middle term is +2xy. Memory cue: x^2, two xy tiles, and y^2 make a positive square (x+y)^2.

The correct answer is A: \((x+2)^2=x^2+4x+4\). Begin with the areas in the model. The large \(x\)-by-\(x\) square has area \(x^2\). The two strips have areas \(2x\) and \(2x\), so together they contribute \(4x\). The four unit squares contribute \(4\). The pieces form one larger square whose side is \(x+2\), so its area is \((x+2)^2\). Expanding confirms that \((x+2)^2=x^2+4x+4\). Option B has a negative middle term and would describe subtraction strips. Option C is a difference-of-squares identity with no middle strips. Option D has only \(2x\), not the required \(4x\), and is not the area of the complete square. Memory cue: for \((x+c)^2\), use \(x^2+2cx+c^2\).

The correct answer is A. An area model represents a multiplication by dividing a rectangle or square into smaller regions. The area of each region is found by multiplying its length by its breadth, and the total area is obtained by adding the regions. For example, a rectangle with sides \(x+2\) and \(x+3\) gives regions \(x^2\), \(3x\), \(2x\), and \(6\); their sum represents the product. Option B is wrong because division is not the defining operation of an area model. Option C is wrong because plus and minus signs determine whether regions are added or subtracted, especially in identities involving \(x-2\). Option D is wrong because the model keeps and combines all relevant terms. Memory cue: split the sides, multiply each pair of side parts, then add signed regions.

The correct answer is B: \(16\). In the model of \((m+4)^2\), one side is split into lengths \(m\) and \(4\), and the other side is split in the same way. The small corner has side length \(4\) on both directions, so its area is \(4\times4=4^2=16\). The \(m\)-by-\(m\) region has area \(m^2\), while the two mixed rectangles together have area \(4m+4m=8m\). Option A describes the variable square, not the constant corner. Option C describes only one mixed rectangle. Option D is the combined area of the two mixed rectangles, not the small square. Therefore B is the only answer that matches the constant corner. Memory cue: in \((m+c)^2\), the corner always contributes \(c^2\).

The correct answer is C: \(10\). In an area model for \((a+2)(a+5)\), one side is divided into lengths \(a\) and \(2\), while the other is divided into \(a\) and \(5\). The constant corner is formed by the two constant lengths, 2 and 5. Its area is therefore \(2\times5=10\). Full expansion gives \((a+2)(a+5)=a^2+5a+2a+10=a^2+7a+10\), confirming that the constant term is 10. Option A is only one side length, not an area. Option B is the other side length, also not an area. Option D is \(2+5\), which is a sum rather than the corner area. Memory cue: the constant term in a product of binomials comes from multiplying the two constant parts.

Expand using the area-model idea: \((x+1)(x+4)=x(x+4)+1(x+4)=x^2+4x+x+4=x^2+5x+4\). Hence the middle \(x\)-term is \(5x\).

The correct answer is B. The area of a square is side times side, so a square with side \(u-v\) has area \((u-v)^2\). Apply the identity \((p-q)^2=p^2-2pq+q^2\), with \(p=u\) and \(q=v\). This gives \((u-v)^2=u^2-2uv+v^2\). The negative middle term appears because the two cross-regions are subtracted in the visual arrangement. Option A, \(u^2-v^2\), is the difference of squares and factors as \((u+v)(u-v)\); it is not the square of \(u-v\). Option C is the identity for \((u+v)^2\). Option D does not follow from squaring a binomial. Memory cue: a minus inside a square produces a negative middle term, but the final square term remains positive.

The correct answer is C: \(200\). Use the difference-of-squares identity \(A^2-B^2=(A+B)(A-B)\). Here \(A=15\) and \(B=5\), so \(15^2-5^2=(15+5)(15-5)=20 imes10=200\). In the rectangle model, the two side lengths are therefore 20 and 10, and their product is the area, 200 square units. Option A is incorrect because it does not equal the given expression. Option B is not obtained from either factor or direct subtraction. Option C correctly uses both the sum and the difference factors. Option D also has no valid step leading from the expression. One may check directly: \(15^2=225\), \(5^2=25\), and \(225-25=200\). Memory cue: difference of squares means add for one side and subtract for the other.

The correct answer is D: a square of area \(16\). The term \(8x\) can be split into two equal rectangles, each with area \(4x\). These rectangles fit along the sides of an \(x\)-by-\(x\) square, leaving a corner whose side length is 4. To fill that corner, add a square with area \(4^2=16\). The completed model represents \(x^2+8x+16=(x+4)^2\). Option A adds 8, which is the coefficient of \(x\), not the missing square area. Option B has no matching half-coefficient square. Option C squares 8 instead of half of 8 and is too large. Option D correctly follows the complete-square rule: square half the coefficient of the linear term. Memory cue: for \(x^2+bx\), add \((b/2)^2\).

The correct answer is A: a square divided into three parts. Divide each side of a large square into lengths \(a\), \(b\), and \(c\). The area model contains three diagonal squares with areas \(a^2\), \(b^2\), and \(c^2\). For each pair of different lengths, there are two matching rectangles: two \(a\)-by-\(b\) rectangles give \(2ab\), two \(b\)-by-\(c\) rectangles give \(2bc\), and two \(c\)-by-\(a\) rectangles give \(2ca\). Adding all regions gives \((a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\). Option B has no two-dimensional areas, option C does not divide the square into these paired rectangles, and option D does not represent the required square expansion. Memory cue: every pair appears twice as two opposite rectangular regions.

The correct answer is B: \(361\). A square's area is the square of its side, so the required area is \((20-1)^2\). Apply \((a-b)^2=a^2-2ab+b^2\): \((20-1)^2=20^2-2(20)(1)+1^2=400-40+1=361\). Direct checking gives \(19^2=361\). Option A, 399, comes from incorrectly doing only \(400-1\); the middle term \(-2ab\) cannot be omitted. Option C, 381, does not follow from the identity or direct squaring. Option D, 421, uses the wrong sign or adds instead of subtracting the cross-term. The visual model also shows why the area is not simply the large square minus one unit: two rectangular strips must be removed, and the corner unit must be restored. Memory cue: in \((a-b)^2\), subtract twice the product and then add \(b^2\).

The correct answer is C. The difference-of-squares identity is (a + b)(a − b) = a² − b². In this rectangle model, the two factors have different signs in their algebraic parts, so there is no pair of positive ab rectangles producing a 2ab term. For A, a square of side a + b has area (a + b)² = a² + 2ab + b², so 2ab appears. For B, a square of side a − b has area a² − 2ab + b², so a middle term still appears, although it is negative. D explicitly contains two ab rectangles, whose combined area is 2ab. Therefore only C has no 2ab term. Memory cue: the difference of squares has only two square terms because the mixed terms cancel.

The correct answer is B: 12x. Imagine the square of side x + 6 divided into four parts: an x by x square, two equal rectangles, and a 6 by 6 square. Each of the two equal rectangles has one side x and the other side 6, so each area is x × 6 = 6x. Since there are two such rectangles, their total area is 6x + 6x = 12x. A gives the area of only one rectangle, not both. C multiplies 36 by x and does not describe either rectangle; 36 is the area of the small corner square. D is the area of the x by x square, not the combined rectangle area. The complete expansion is (x + 6)² = x² + 12x + 36, which confirms the middle term.

C is correct. Since x² + 14x + 49 = (x + 7)², the area is the square of a side of length x + 7. This follows by expanding (x + 7)²: x² + 14x + 49. For a square model with positive side lengths, the side is x + 7.

The correct answer is B: the model of (a + b)². In a visual square model, an a by a square, two a by b rectangles, and a b by b square can be placed together without gaps. The outside length in each direction is a + b, so the whole figure is a square with area (a + b)². Adding the four regions gives a² + ab + ab + b² = a² + 2ab + b². A represents a square of difference, whose outside side is a − b and whose mixed regions are subtracted. C represents a rectangle with sides a + b and a − b, not a larger square. D represents only one rectangular region with area ab. Thus B is the only model whose outer boundary has side a + b. Memory cue: read the outside boundary first; its side tells you which square identity is being modelled.

C is correct because an area model shows how the parts of an expression combine. For example, a square with side p + q can be split into regions with areas p², pq, pq, and q², giving p² + 2pq + q². The model connects the symbols in an identity to visible areas; multiplication is still used to find each area.

The direct answer is B: 3×8, which equals 24. The two side lengths are s+3 and s+8. In the area model, the constant region is the small corner where the two constant lengths meet. Its dimensions are 3 and 8, so its area is 3×8=24. The complete multiplication would be (s+3)(s+8)=s^2+8s+3s+24=s^2+11s+24. Option A adds the constants rather than finding the corner area. Option C describes a variable rectangle with area 3s, so it is not constant because it changes when s changes. Option D similarly has area 8s and also depends on s. Only B contains no variable and correctly uses multiplication, because area is length times breadth. Memory cue: in a binomial area model, the constant term comes from multiplying the two constant terms.