Visual models of identities

The direct answer is C. Expand the larger square model: (a+b)^2=a^2+2ab+b^2. The second model has area a^2+b^2. Subtracting gives [(a+b)^2]-(a^2+b^2)=(a^2+2ab+b^2)-(a^2+b^2)=2ab. In the visual picture, the larger square contains the a^2 square, the b^2 square, and two mixed rectangles, each of area ab. Removing the two square regions leaves those two rectangles, so their combined area is 2ab. A counts only one mixed rectangle. B is a length-like sum and does not represent the missing rectangular areas. D subtracts squares incorrectly and is not the difference of the two given models. Therefore C is correct. Memory cue: the extra part in (a+b)^2 beyond a^2+b^2 is exactly the two cross-rectangles, 2ab.

The direct answer is B: the other side is a + b. The original square has side a + b, so its area is (a + b)². Cutting and rearranging the pieces does not change the total area. The new rectangle has one side a + b, so let its other side be L. Its area is (a + b)L. Because the area is unchanged, (a + b)L = (a + b)². For a valid positive side length, divide by a + b to obtain L = a + b. A, a − b, would generally produce a different area and may not even be positive for all values. C, 2a + 2b, would make the rectangle twice as large when multiplied by the unchanged side. D, ab, has the wrong form and does not preserve the original area. The key idea is conservation of area: rearrangement changes shape, not total area. Memory cue: same area plus one same side forces the other side to remain the same.

The direct answer is C: x = 5. A square with side x + 3 has area (x + 3)² = x² + 6x + 9. The model already identifies the x² part and the 9 part as colored. Therefore, the uncolored regions together represent the middle term 6x. Since that uncolored area is 30, set 6x = 30. Dividing both sides by 6 gives x = 5. A, x = 3, would make the uncolored area 18, not 30. B, x = 4, would make it 24. D, x = 6, would make it 36. Only C satisfies the area condition. It is important not to set x² + 9 equal to 30, because those are the colored parts, not the uncolored strips. Memory cue: in (x + 3)², the two cross-strips together are 3x + 3x = 6x.

The direct answer is D: it is always a perfect-square model with side x + k. Use the identity (x + k)² = x² + 2xk + k², and 2xk is the same as 2kx. Visually, a square of side x + k can be divided into an x-by-x square of area x², two rectangles of area xk each, and a k-by-k square of area k². Their total is x² + xk + xk + k² = x² + 2kx + k². A is wrong because the expression is explicitly a square identity. B is wrong because k = 0 is only one special case, not a requirement. C is wrong because x and k need not be equal. D correctly states both the general condition and the side length. The word always assumes values for which the displayed lengths form a valid geometric model. Memory cue: first term square, last term square, and middle term twice their product.

The direct answer is A: 169. A square with side \(a+b\) is partitioned into four non-overlapping pieces: the square of area \(a^2\), the square of area \(b^2\), and two rectangles whose combined area is \(2ab\). The problem already gives the first two pieces together as 85 and the rectangular pieces together as 84. Since the whole area is the sum of all pieces, calculate \(85+84=169\). Option B, 1, is the difference between the given totals, not the area of the complete model. Option C counts only the two corner squares and leaves out the rectangles. Option D counts only the rectangles and leaves out the corner squares. The model uses addition because the pieces fill the square without overlap. Memory cue: total area of a partition equals the sum of every non-overlapping part.

The direct answer is C: two rectangles, each measuring \(a\times b\). Imagine a square whose side is split into lengths \(a\) and \(b\) in both directions. The partition creates one \(a\times a\) square with area \(a^2\), one \(b\times b\) square with area \(b^2\), and two remaining rectangles. Each remaining rectangle has dimensions \(a\) and \(b\), so each area is \(ab\). Together they contribute \(ab+ab=2ab\). Option A represents only the \(a^2\) corner square. Option B represents only the \(b^2\) corner square. Option D describes the whole square itself, whose area is the entire left-hand side, not just the middle part. Memory cue: the coefficient 2 means that two identical \(ab\) rectangles are present.

The direct answer is A. The whole figure is a square whose side is p+q, so its total area is (p+q)². The parts have areas p², pq, pq and q². Adding them gives p²+pq+pq+q²=p²+2pq+q². Therefore (p+q)²=p²+2pq+q². B describes the square of a difference; its two cross-regions would contribute a negative middle term in an algebraic representation. C describes a difference of squares and would involve a product of two different side lengths, not one complete square divided into four positive regions. D misses one of the two pq rectangles. The visual count is the key: two equal pq regions always combine to 2pq. Memory cue: square of a sum means first square, twice the product, last square.

The direct answer is A. A square with side x+5 is divided into an x by x square, two rectangles of dimensions x by 5, and a 5 by 5 square. Their areas are x², 5x, 5x and 25. Thus (x+5)²=x²+5x+5x+25=x²+10x+25, not merely x²+25. Option A is the model that visibly displays the two missing middle regions, so it catches the common error of squaring each term separately and forgetting the cross terms. B shows only one part and cannot represent the complete square. C uses x−5, which belongs to a different identity. D represents (x+5)(x−5)=x²−25, the difference-of-squares situation, not (x+5)². Memory cue: when adding two lengths inside a square, two cross-rectangles must appear.

The direct answer is A. Since the side of the complete square is made from p and q, the total side is p+q. Therefore the whole area is (p+q)². The four regions add to p²+pq+pq+q², and the two like terms combine to give p²+2pq+q². This proves (p+q)²=p²+2pq+q². B has the correct square-shaped pattern but applies to a difference and therefore has a negative middle term. C is the difference-of-squares identity and does not describe four positive regions inside one square of side p+q. D is a cubic expansion, which would involve volume-like terms and four terms of different degrees, not four areas. Memory cue: count the two pq regions; they are the source of the positive 2pq middle term.

The direct answer is A. The two square pieces contribute a² and b². The two equal rectangles each have area ab, so together they contribute 2ab. Adding all four regions gives a²+ab+ab+b²=a²+2ab+b². The outside side is formed by adding the corresponding lengths a and b, so the whole figure has area (a+b)². Therefore the model represents (a+b)²=a²+2ab+b². B would require negative cross terms, associated with a difference. C represents multiplying a sum and a difference, not adding the four positive parts of one square. D is the cube of a binomial and is not an area decomposition into these four quadratic regions. Memory cue: two ab rectangles mean a positive 2ab middle term.

The direct answer is option A: 14. Recognize the perfect-square identity m² + 2mn + n² = (m+n)². The given equation therefore becomes (m+n)² = 196. Since 196 = 14², taking square roots gives m+n = 14 or m+n = −14. The question says that m+n is the positive outer side of the square, so the positive value 14 must be selected. Option B, 98, is half of 196 and is not its square root. Option C, 28, is twice the positive square root, so it does not satisfy the equation. Option D, 196, is the area represented by the squared expression, not the side length. The important idea is that a square equation normally permits both signs, but a geometrical length or explicitly positive side uses the positive root. Memory cue: match m²+2mn+n² with (m+n)², then take the positive square root for a side.

The direct answer is option A. The entire square has side length x+y because the original side is split into adjacent parts x and y. Therefore its total area is (x+y)². The partition contains one x-by-x square with area x², one y-by-y square with area y², and two crosswise rectangles, each with area xy. Adding the pieces gives x² + xy + xy + y² = x² + 2xy + y². Option B describes a model with two negative cross-rectangles, which would correspond to (x−y)², not the stated positive pieces. Options C and D concern products of a sum and a difference and do not match two positive xy rectangles. In an area model, count how many xy rectangles appear and inspect their signs. Two positive rectangles mean +2xy. Memory cue: the square of a sum has positive middle term; the square of a difference has negative middle term.

The direct answer is option A: 72. The question asks for the difference between the two given square areas, so subtract the smaller area from the larger area: 121 − 49 = 72 square units. The visual identity behind this idea is a² − b² = (a+b)(a−b). Here the areas themselves are already supplied, so no square roots are needed for this particular calculation. Option B, 84, does not result from subtracting the two areas; it may arise from an unrelated combination of numbers. Option C, 170, is their sum, 121 + 49, not their difference. Option D, 35, is the difference between the square roots 11 and 7, not the difference of the square areas. Read the wording carefully: it asks for the area difference, not the difference of side lengths. Memory cue: area difference means subtract 121 and 49 directly; side difference would involve 11 and 7 instead.

The direct answer is option C: 96. The given square areas allow us to find the positive side lengths because a and b are stated to be positive: a = √64 = 8 and b = √36 = 6. The model of (a+b)² contains two rectangles, each with area ab = 8×6 = 48. Their combined area is 2ab = 2×48 = 96. Option A, 100, is a common error from adding 64 and 36, which gives the two corner-square areas together but not the rectangular middle area. Option B is only ab, one rectangle, not both. Option D, 28, is the difference between the side lengths or between other given values and does not represent 2ab. The positivity condition matters because square roots could otherwise have signs, but the visual side lengths are taken positive. Memory cue: square areas first give side lengths; then multiply the two sides and double the result.

The direct answer is B: 2mn. A square with side m+n has total area (m+n)². In the four-part model, one m by m square has area m² and one n by n square has area n². The other two regions are rectangles with dimensions m and n, so each rectangle has area mn. Since there are two such rectangles, their combined area is mn+mn=2mn. This gives the identity (m+n)²=m²+2mn+n². A is only the area of one rectangle, not both. C is a length, not an area, and it also lacks the required product of dimensions. D is the sum of the two square regions and does not represent the remaining parts. Memory cue: in a binomial square, the cross term appears twice because the two mixed rectangles are equal.

The direct answer is B: x+6. Compare the expression with the perfect-square identity (x+a)²=x²+2ax+a². Here the constant term is 36, which is 6², so the added side length is 6. The middle term predicted by this value is 2·x·6=12x, exactly the given middle term. Therefore x²+12x+36=(x+6)², and the side of the square is x+6. A would produce (x+12)²=x²+24x+144, so it is much too large. C would produce x²−12x+36 and has the wrong sign in the middle term. D is not the square of a binomial and represents an unsuitable algebraic form for a side. Memory cue: take the square root of the constant term, then check twice that number against the middle coefficient.

The correct answer is A: \((a+b)^2=a^2+2ab+b^2\). Imagine a square whose side is made from lengths \(a\) and \(b\), so its full side is \(a+b\). Its total area is therefore \((a+b)^2\). The partition creates one square of area \(a^2\), one square of area \(b^2\), and two congruent rectangles. Each rectangle has length \(a\) and width \(b\), so each has area \(ab\); together they have area \(2ab\). Adding all four regions gives \(a^2+2ab+b^2\). B would require subtraction strips and a negative middle term. C describes two squares whose difference is rearranged into a rectangle, not this four-part large square. D incorrectly omits the two rectangle areas; in general, \(a^2+b^2\) is not equal to \((a+b)^2\). Memory cue: two equal cross-rectangles mean \(2ab\), while two subtraction strips usually signal \(-2ab\).

The correct answer is C: the combined area of the cross rectangles is \(2ab\). In the square model for \((a+b)^2\), the regions are an \(a^2\) square, a \(b^2\) square, and two cross rectangles. Each cross rectangle has side lengths \(a\) and \(b\), so each area is \(ab\). Since there are two of them, their combined area is \(ab+ab=2ab\). Adding the corner squares gives \(a^2+2ab+b^2\). A is the combined area of the two square regions, not the cross rectangles. B gives the area of only one cross rectangle and misses the second one. D gives a length expression, not an area, because it has only one dimension. The comparison \((a+b)^2-(a^2+b^2)=2ab\) also confirms the answer. Memory cue: the coefficient 2 counts the two identical rectangles.

The correct answer is A. Divide both sides of the large square, whose side is a + b, into segments of lengths a and b. The horizontal and vertical divisions create four regions. The a-by-a corner is a square of area a^2. The b-by-b corner is a square of area b^2. The two remaining regions each have dimensions a and b, so each is an ab rectangle. Their combined area is 2ab. Adding all four regions gives a^2 + ab + ab + b^2 = a^2 + 2ab + b^2 = (a + b)^2. Option B has two a^2 squares, but only one a-by-a region exists. Option C has two b^2 squares, which is also incorrect. Option D labels the middle regions by their total side length instead of their actual dimensions. Memory cue: a binomial square always has two end squares and two equal cross-rectangles.

The correct answer is B: m = 20. A perfect-square area model with a main side segment x and a corner side segment 10 represents (x + 10)^2. Apply the identity (x + r)^2 = x^2 + 2rx + r^2 with r = 10. This gives (x + 10)^2 = x^2 + 20x + 100. Comparing it with x^2 + mx + 100 shows that m, the coefficient of x, must be 20. The visual explanation is the same: there are two equal side rectangles, each with area 10x, so together they contribute 10x + 10x = 20x. Option A counts only one rectangle. Option C is the corner square’s area, 10 × 10, not the coefficient of x. Option D is twice the correct middle coefficient and has no matching expansion. Memory cue: in (x + r)^2, the middle coefficient is always 2r, while the final constant is r^2.

The correct answer is A. Rewrite both terms as squares: 9x^2 = (3x)^2 and 1 = 1^2. Therefore 9x^2 - 1 is the difference between the areas of a square with side 3x and a square with side 1. The correct visual model removes the 1-by-1 square from the larger 3x-by-3x square. Algebraically, this is (3x)^2 - 1^2 = (3x - 1)(3x + 1). Option B uses a square with side 9x, whose area would be 81x^2, not 9x^2. Option C represents (3x + 1)^2, which includes positive cross terms and is not a difference. Option D represents (3x - 1)^2, which also contains a square term plus a middle term, not 9x^2 - 1. Memory cue: identify the side by taking the square root of each square term; the side of 9x^2 is 3x, not 9x.

The correct answer is B: x²+2x+1. The area of any square is side × side, or the square of its side. Here the side is x+1, so area = (x+1)(x+1) = (x+1)². Multiply each term in the first bracket by each term in the second bracket: x×x=x², x×1=x, 1×x=x, and 1×1=1. Adding these parts gives x²+x+x+1=x²+2x+1. In the visual model, x² is the large x-by-x square, and the two x terms are the two congruent rectangles. Option A misses the two x rectangles. Option C has only one x term, so it misses one rectangle. Option D is 2x+1 and does not include the main x² square. Remember: (x+1)² is not x²+1; the middle term 2x must be included.

The correct answer is C: 4. In an area model for (a+2)², the side is split into two lengths: a and 2. This creates four parts: an a-by-a square with area a², two rectangles with areas 2a each, and a small 2-by-2 corner square. The question asks only for the small corner square, not for the whole large square. Its side is 2, so its area is 2×2=2²=4. Option A is only the side length 2, not its area. Option B, 2a, describes one of the rectangular parts formed by a length a and a length 2. Option D, a², describes the square whose side is a. Therefore C is the only suitable answer. A useful memory cue is that the constant term in (a+2)² comes from the constant-by-constant corner: 2²=4.

The correct answer is A: x²+4x+3. The area of a rectangle is length × breadth, so we calculate (x+3)(x+1). In the area model, split the length into x and 3 and the breadth into x and 1. The four smaller areas are x×x=x², x×1=x, 3×x=3x, and 3×1=3. Add them: x²+x+3x+3=x²+4x+3, because x and 3x are like terms. Option B leaves out the 3x contribution. Option C uses 2x even though the two cross-products are x and 3x, which total 4x. Option D has constant term 4, but the constant corner is 3×1=3, not 4. Thus A is the only complete expansion. Memory cue: for (x+p)(x+q), the middle coefficient comes from px+qx and the constant term is pq.

The correct answer is B: ab. Divide each side of the large square into parts of lengths a and b. The four resulting regions are an a-by-a square, two rectangles, and a b-by-b square. Each of the two rectangles has one side a and the other side b, so each has area a×b=ab. Together, these two rectangles contribute ab+ab=2ab to the identity (a+b)²=a²+2ab+b². Option A is the area of the a-by-a corner square. Option C is the area of the b-by-b corner square. Option D is a length expression, not an area of one of the regions. Therefore B is correct. Be careful about the wording: one rectangle has area ab, while the pair together has total area 2ab.