Visual models of identities

The correct answer is A: (x+y)². The large square has side x+y, so its total area is (x+y)². The four regions have areas x², xy, xy, and y². Adding them gives x²+xy+xy+y²=x²+2xy+y², which is exactly the expansion of (x+y)². Option A is correct because it represents the whole large square. Option B is wrong because (x−y)²=x²−2xy+y² has a negative middle term, whereas the two xy rectangles here are added. Option C is wrong because x²+y² omits both xy rectangles. Option D is wrong because x²−y² represents a difference of squares and includes subtraction, not the four positive regions in the diagram. The model shows that the middle term comes from two equal rectangles, each with area xy. Memory cue: two positive xy pieces always combine to +2xy.

The correct answer is A: 2a and −2a cancel. Multiply the two side expressions using distribution: (a+2)(a−2)=a·a+a(−2)+2a+2(−2)=a²−2a+2a−4. The middle terms −2a and +2a are opposites, so their sum is zero. The result is a²−4, which is the identity (a+b)(a−b)=a²−b² with b=2. Option A is correct because it identifies the exact cancellation. Option B is wrong because a² remains the main square term; it does not become zero. Option C is wrong because the constant term is not added: 2×(−2)=−4, so 4 is subtracted. Option D is wrong because the two side expressions need not be equal for multiplication, and unequal sides do not explain cancellation. In an area model, the +2a and −2a rectangular regions have equal magnitude but opposite signs, so they remove each other algebraically. Memory cue: opposite middle terms in (a+b)(a−b) cancel, leaving a²−b².

The direct answer is option C: 7. A square area model for (x + k)² has three parts: x², two rectangles each measuring kx, and a small corner square of area k². Compare x² + 14x + 49 with x² + 2kx + k². Since k² = 49, k = 7, and 2kx = 2 × 7 × x = 14x, the expression is (x + 7)². Therefore, the small corner square has side 7. Option A, 14, is the coefficient of the middle term, not a side. Option B, 49, is the corner square’s area, not its side. Option C is correct because 7² = 49. Option D, 21, gives neither the required constant area nor the required middle term. Memory cue: in a perfect square, the small-corner side is the square root of the constant term; always distinguish area from side length.

The direct answer is option A. The area of a square equals the square of its side. A large square with side s has area s². A small square with side 5 has area 5² = 25. Subtracting the small area from the large area gives s² − 25, exactly the expression in the question. Option A therefore describes the correct model. Option B would give s² − 25² = s² − 625, because 25 is being treated as a side rather than as an area. Option C names two lengths whose product is (s + 5)(s − 5) = s² − 25, but it does not describe the difference of the areas of squares with those sides. Option D would give (5s)² − s² = 24s². Memory cue: in a difference-of-squares area model, square each stated side before subtracting.

The direct answer is option C: 4. Split the side 3x + 2 into two parts, 3x and 2, in both directions. The square then contains a large square of area (3x)² = 9x², two rectangles each of area (3x)(2) = 6x, and one small corner square with side 2. Its area is 2² = 4. Option A, 3x, represents a length part, not an area. Option B, 6x, is the area of one of the two rectangular strips. Option C is correct because the small corner has side 2 and area 4. Option D, 9x², is the area of the large square formed by the 3x portions. The whole square has area (3x + 2)², but the question asks for only one component. Memory cue: the constant part of a square-side expression forms the corner square, so square that constant.

The direct answer is option A: (x + 3)(x + 2). In a rectangular area model, adjoining lengths multiply to make each region. The x² region comes from x by x. The 3x region comes from x by 3, and the 2x region comes from x by 2. The remaining corner is 3 by 2, giving 6. Therefore, the two outside side lengths are x + 3 and x + 2, so the factorised expression is (x + 3)(x + 2). Expanding confirms it: x² + 2x + 3x + 6 = x² + 5x + 6. Option B would produce x² + 7x + 6, not the given parts. Option C would have two 3x rectangles and a 9 corner. Option D would have two 2x rectangles and a 4 corner. Read the strip widths along the two sides rather than guessing from the constant term alone.

The direct answer is option B: w + 3. Use the identity (a + b)² = a² + 2ab + b². Here the first term w² shows that one part of the side is w, and the constant term 9 = 3² shows that the other part is 3. The middle term must then be 2 × w × 3 = 6w, which matches the expression. Hence w² + 6w + 9 = (w + 3)², so the outer side is w + 3. Option A would give (w + 6)² = w² + 12w + 36. Option C would give a negative middle term, w² − 6w + 9. Option D would give a much larger constant and middle term. A useful check is to square every proposed side and compare all three terms, not just the last term.

The direct answer is option A: 2a² + a. The area of a rectangle is length multiplied by breadth. Therefore, area = (2a + 1) × a. Distribute a to every term inside the bracket: a × 2a = 2a² and a × 1 = a. Adding these parts gives 2a² + a. In the visual model, the length 2a + 1 is split into strips 2a and 1; their areas with breadth a are 2a² and a. Option B omits the factor a in the second part and incorrectly writes 1 instead of a. Option C adds lengths instead of multiplying them, so it is not an area. Option D has the terms and coefficients in the wrong arrangement; a × 2a gives 2a², not 2a. Memory cue: for a rectangle, multiply the breadth by every separate piece of the length.

The direct answer is option B: −4x. Write the side as x + (−2). Using (a + b)² = a² + 2ab + b², we get (x − 2)² = x² + 2x(−2) + (−2)² = x² − 4x + 4. The middle part consists of two rectangular products, each x(−2) = −2x, so together they give −4x. Option A, 4x, has the wrong sign; subtracting 2 makes the cross-term negative. Option B is correct because it includes both strips and their negative sign. Option C, 2x, is only the magnitude of one strip and is incomplete. Option D, −2x, is one strip only, not the total of both. The small corner contributes (−2)² = 4 and must not be confused with the middle part. Memory cue: (x − b)² has middle term −2bx.

The direct answer is option A: 77. First find the area of the large square: 9² = 81. Then find the area of the removed small square: 2² = 4. The remaining area is the large area minus the small area, so 81 − 4 = 77. This is the numerical form of the difference-of-squares expression 9² − 2². Option A is therefore correct. Option B, 81, is only the area of the large square and ignores the removed part. Option C, 79, results from subtracting 2 rather than subtracting the small square’s area 4. Option D, 85, adds the two square areas instead of finding the remaining region. A safe method is to translate each side into an area first and subtract only after both squares have been calculated.

The direct answer is option A. The x² region is a square with side x, and the y² region is a square with side y. The two remaining regions are rectangles, each with area xy. Together they contribute xy + xy = 2xy. Therefore, the total area is x² + 2xy + y². Since the four pieces form one large square whose side is x + y, its area is (x + y)². Thus x² + 2xy + y² = (x + y)². Option B is the difference-of-squares identity and has no two positive xy rectangles. Option C would require negative middle regions. Option D incorrectly leaves out the two xy regions; in general, x² + y² is not equal to (x + y)². Memory cue: two positive cross-rectangles mean a positive 2xy middle term.

The direct answer is option C: a² − b². Rearranging or cutting and repositioning pieces can change the shape of a figure, but it does not change the total area, provided no piece is lost or overlapped. The original model represents the area of a large square, a², minus the area of a smaller square, b². Therefore, its remaining area is a² − b². When the pieces are rearranged into a rectangle, the same area is also expressed through the identity a² − b² = (a − b)(a + b), but the question asks what the area remains equal to, so a² − b² is the direct answer. Option A adds the two square areas instead of subtracting. Option B is only the product of the two variables doubled and is not generally equal to the remaining area. Option D is a length expression, not an area. Memory cue: rearrangement changes shape, not area.

Correct answer: A. Divide each side of the large square into lengths a and b. This creates four regions: an a-by-a square with area a², a b-by-b square with area b², and two rectangles. Each rectangle has length a and width b, so each has area ab. Adding the two equal rectangular regions gives ab+ab=2ab. Therefore, the two rectangles together represent the middle term 2ab in (a+b)²=a²+2ab+b². Option B represents only a², and option C represents only b². Option D represents the entire area, namely (a+b)², not just its middle part. Memory cue: the coefficient 2 appears because there are two identical ab rectangles.

Correct answer: A. The two factors are identical: (x+5) and (x+5). A rectangle whose length and width are equal is a square, so the model is a square with side x+5. Its area can be written as (x+5)², and expanding it gives x²+10x+25. In an area diagram, this would contain an x-by-x square, two 5-by-x rectangles, and a 5-by-5 square. Option B is incorrect because a difference-of-squares model has factors such as (x+5)(x-5), not two equal positive factors. Option C is not an area model, and option D would involve triangular regions or a triangular shape. Memory cue: equal factors or equal sides signal a square model.

Correct answer: B. Start with the square of area x². Removing two strips, each with area 4x, removes 8x altogether. Adding the corner square of area 16 gives x²−8x+16. Now use the identity (a−b)²=a²−2ab+b². Taking a=x and b=4 gives (x−4)²=x²−2(x)(4)+4²=x²−8x+16. Thus the completed figure is a square with side x−4. Option A would expand to x²+8x+16, requiring addition rather than removal of the strips. Option C omits the two strip terms, and option D has the wrong degree and does not describe completing a square. Memory cue: remove two 4x strips and add 4² means subtract 8x and complete (x−4)².

Correct answer: A. Split one side into x and 3, and the other side into x and 4. The four small regions then have areas x·x=x², x·4=4x, 3·x=3x, and 3·4=12. The two regions that contain one x and one constant are therefore 3x and 4x. Their combined contribution is 3x+4x=7x, but 7x is the sum of the two regions, not the names of the two separate parts. Option B lists the two diagonal corner regions, not the two x-containing mixed rectangles. Option C mixes a sum with a separate area, and D merely repeats side lengths. Memory cue: multiply the labels on the crossing sides of each small rectangle.

Correct answer: C. The expression contains one a² square, two ab rectangles, and one b² square. These four regions fit together to form one complete square whose side is a+b. Therefore its total area is (a+b)², and the identity is (a+b)²=a²+2ab+b². Option A is the square-of-a-difference identity and would contain a negative middle term, a²−2ab+b². Option B is the difference of two squares and has only two terms, so it cannot match four positive regions. Option D gives only one ab rectangle, not the whole model. Memory cue: two positive ab parts mean a plus b is the outer side.

Correct answer: B. Compare the expression with the identity (p+q)²=p²+2pq+q². Here p=x and q=y, so x²+2xy+y²=(x+y)². A square area is the square of its outer side; therefore the side of the model is x+y. Visually, the model has an x-by-x square, two x-by-y rectangles, and a y-by-y square. Option A would represent (x−y)² and would have −2xy. Option C would produce (2x+y)²=4x²+4xy+y², while option D would produce x²+4xy+4y²; neither matches the given expression. Memory cue: positive two-times-product means add the two side labels.

Correct answer: A. The full side is made by joining lengths x and y, so it is x+y. The whole square therefore has area (x+y)². Its four regions have areas x², xy, xy, and y². Adding them gives x²+xy+xy+y²=x²+2xy+y², proving the identity in option A. Option B describes a difference of lengths and would require negative middle terms; the diagram has two positive xy regions. Option C describes a product of conjugate factors and has no two positive xy rectangles. Option D incorrectly leaves out the two cross regions. Memory cue: count both identical cross-rectangles; they combine to 2xy.

Correct answer: C. The area of a square is side multiplied by itself, so the required area is (g−1)². Apply (a−b)²=a²−2ab+b² with a=g and b=1: (g−1)²=g²−2(g)(1)+1²=g²−2g+1. Therefore option C is correct. Option A, g²−1, is the difference of squares g²−1² and is not the same as the square of a difference; it misses the middle term. Option B belongs to (g+1)² and has a positive middle term. Option D has an incorrect coefficient and does not result from multiplying (g−1) by itself. Memory cue: when a binomial is squared, never omit the middle term.

The correct answer is B: \(4a^2\). In the visual area model, the side \(2a+3\) is divided into two parts, \(2a\) and \(3\). The large square is formed by using the \(2a\) part along both its length and breadth. Therefore, its side is \(2a\), so its area is \((2a)(2a)=(2a)^2=4a^2\). Notice that both the numerical coefficient and the variable are squared: \(2^2\times a^2=4a^2\). Option A, \(2a^2\), forgets to square the coefficient 2. Option C, \(6a\), is obtained from \((2a)(3)\), so it represents one rectangular cross-part, not the large square. Option D, \(9\), is the area of the small square with side 3. A useful memory cue is: for any square region, multiply its side by itself; never square only the letter and leave the coefficient unchanged.

Correct answer: A. The outer side is explicitly x+10, so the complete square has area (x+10)². In its area model, the small corner is formed by the 10-by-10 lengths, giving area 10²=100. Expanding the complete model would give x²+20x+100: one x² square, two 10x rectangles, and the 100 corner square. Option B has outer side x−10, not x+10. Option C is a difference of two square areas and does not describe the complete outer square. Option D is only the area of one mixed rectangle, 10 times x, not the whole model. Memory cue: the sign in the outer side controls the middle term, while the corner area is the square of 10.

The correct answer is B: 3. In the visual model of x^2 - 9, the large square has area x^2 and the removed small square has area 9. The side of a square is the positive square root of its area, because area = side × side = side^2. Since 9 = 3^2, the side of the removed square is √9 = 3. Option A, 9, is the area of the small square, not its side. Option B is correct because 3 × 3 = 9. Option C, x - 3, is a possible length used after rearranging a difference-of-squares model, but it is not the side of the removed square whose area is 9. Option D, x + 3, is the other factor in x^2 - 9 = (x - 3)(x + 3), not the side of the removed square. Memory cue: when an area is given for a square, take its square root to find the side.

The correct answer is B: 10x. The corner regions are squares with areas x^2 and 25. Their side lengths are x and 5, because x^2 = x × x and 25 = 5 × 5. A square with side x + 5 has area (x + 5)^2 = x^2 + 2(x)(5) + 25 = x^2 + 10x + 25. In a visual model, the middle consists of two equal rectangles. Each rectangle has dimensions x and 5, so each has area 5x; together they have area 5x + 5x = 10x. Option A gives only one rectangle, not both. Option B is the complete middle term. Option C incorrectly uses 25 rather than its side 5. Option D doubles the required total again. A useful method is: take the square roots of the two corner areas and double their product. The identity is A^2 + 2AB + B^2.

The correct answer is A: h^2 - 36. The area of a rectangle is length multiplied by breadth, so the required expression is (h + 6)(h - 6). These are conjugate binomials: the first terms are the same and the second terms have opposite signs. Therefore we use (a + b)(a - b) = a^2 - b^2. Taking a = h and b = 6 gives h^2 - 6^2 = h^2 - 36. Option A follows directly. Option B has the wrong sign; this product is a difference of squares, not a sum. Option C is the expansion of (h - 6)^2, which is a different expression because both factors would then have minus signs. Option D is the expansion of (h + 6)^2, also a different product. Expanding directly confirms the result: h^2 - 6h + 6h - 36 = h^2 - 36. Memory cue: matching binomials with opposite signs cancel their middle terms.