Visual models of identities

The correct answer is A. The expression x^2+6x+9 can be split as x^2+3x+3x+9 because the middle term 6x is the total area of two equal 3x rectangles. In a visual square model, x^2 is one square, the two 3x pieces are side strips, and 9 is a 3-by-3 corner square. These four pieces fit together to form a square whose side is x+3. Algebraically, this is the identity x^2+6x+9=x^2+2(3x)+3^2=(x+3)^2. Option B removes an essential area and cannot make the complete square. Option C hides the two separate strips, which is less useful for showing the square. Option D changes the required model and does not explain the identity. Remember: splitting the middle term often reveals the two matching cross-rectangles.

The correct answer is A. The identity (a+b)^2=a^2+2ab+b^2 is represented by one a^2 square, one b^2 square, and two equal rectangles, each with area ab. If only a^2 and b^2 are kept, the two cross-shaped rectangular regions are absent. Their combined area is ab+ab=2ab, and without them the pieces cannot form the complete square with side a+b. Option B is not an area piece in this model; it describes a length rather than a missing region. Option C belongs to a different signed or difference expression and has the wrong dimensions for the missing pair. Option D repeats b^2 squares, but the model needs one b^2 corner square and two ab rectangles. Memory cue: in (a+b)^2, the middle term 2ab comes from two equal rectangles.

The correct answer is C. Apply the identity (p-q)^2=p^2-2pq+q^2 with p=2a and q=b. The first square is (2a)^2=4a^2, the two cross-rectangles together contribute -2(2a)(b)=-4ab, and the small square from the b side has area (-b)^2=b^2. A square area is positive because multiplying a quantity by itself removes its sign. Option A is also positive, but it is the first large square, not the final b-part being asked about. Option B is the negative middle contribution. Option D incorrectly keeps a negative sign on b^2. Common confusion: the minus sign affects the cross term, but the square of the subtracted term is positive.

The correct answer is B. Divide the square with side x+y+z into strips labelled x, y, and z along both directions. The region formed by an x strip crossing a y strip has area xy. There are two such crossings: the x-by-y region and the y-by-x region. Although their numerical areas are equal, they are two distinct rectangles in the visual model. Thus the total contribution of the xy pair is xy+xy=2xy. Option A counts only one orientation and misses the second rectangle. Option C incorrectly counts the z strip as another xy region. Option D has no matching pair of x and y strips. The full identity is (x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2zx, so every different pair contributes twice.

The correct answer is C. In the visual square for (a+b+c)^2, the a and b strips meet in two separate rectangular regions. One has area a times b=ab, and the other has area b times a=ab. Adding them gives ab+ab=2ab. The c-related rectangles contribute 2ac and 2bc, but they do not change the ab contribution. Option A counts only one of the two rectangles. Option B squares both variables and therefore has the wrong type and degree; the region has side lengths a and b, not a^2 and b^2. Option D counts three rectangles even though only two orientations involve a and b. Memory cue: in a square of three summed parts, every distinct pair appears twice, producing 2ab, 2ac, or 2bc.

The direct answer is A: the outer side is (7x+2). A square area is found by multiplying its side by itself, so a perfect-square expression has the form (p+q)^2=p^2+2pq+q^2. Here, 49x^2=(7x)^2 and 4=2^2. The middle term is 2(7x)(2)=28x, so every part matches and 49x^2+28x+4=(7x+2)^2. In the visual model, one side contains a length 7x part and a length 2 part; adding them gives 7x+2. B incorrectly uses 49 instead of its square root, 7. C uses 4 instead of its square root, 2. D doubles 7x without any reason. Memory cue: take the square roots of the first and last terms, then check the middle term.

The direct answer is A: 4x²+11x+6. In a rectangle model, the total area equals the sum of the smaller rectangular areas. Split the sides into 4x and 3, and into x and 2. The four products are (4x)(x)=4x², (4x)(2)=8x, (3)(x)=3x, and (3)(2)=6. Adding them gives 4x²+8x+3x+6. Since 8x and 3x are like terms, they combine to 11x, giving 4x²+11x+6. B omits the 3x rectangle. C omits or miscomputes several products. D changes the leading coefficient 4 to 5. Memory cue: every term in one bracket must meet every term in the other bracket.

The direct answer is A: the model is a square with side (x+6). To complete a square, compare x²+12x+36 with (x+k)^2=x²+2kx+k². The middle coefficient gives 2k=12, so k=6. The constant term confirms this because k²=6²=36. Visually, the x² tile is one large square, and the 12x area is split into two equal rectangles, each 6x. The missing corner is therefore a 6-by-6 square. B would create x²+24x+144. C gives (x+3)(x+9)=x²+12x+27, not 36. D gives (x−6)(x+6)=x²−36. Memory cue: halve the middle coefficient to find the side of the small corner square.

The direct answer is A: the two middle 6x rectangles are missing. A complete square with side x+6 has area (x+6)^2=x²+12x+36. The x² tile supplies the large corner, and the 36 tile supplies the small 6-by-6 corner. However, the two remaining strips must have areas 6x and 6x; together they supply the middle term 12x. If those rectangles are absent, the pieces have total area only x²+36, which is not generally equal to x²+12x+36 and cannot make the full square. B is false because 36=6² is a square. C is false because x² can represent a square area. D is false because x is a variable, not always 6. Memory cue: a square identity needs both equal cross rectangles, not just the two corner squares.

The direct answer is C: the side is (5a−3b). Use (p−q)^2=p²−2pq+q². The first term 25a² is (5a)², and the last term 9b² is (3b)². The middle term from (5a−3b)^2 is −2(5a)(3b)=−30ab, exactly as required. Therefore the whole area is (5a−3b)^2. A has a plus sign and would give +30ab. B uses 25a and 9b instead of their square-root factors and therefore has the wrong scale. D doubles both correct side parts, so its square would be four times the expression. Memory cue: a negative cross term means the two square-root parts are subtracted.

The direct answer is D: there are 6 rectangular cross parts. Imagine a square whose side is split into a, b, and c. In the area model, the three diagonal square parts are a², b², and c². For each pair of different lengths, there are two rectangles: one pair gives ab and ba, another gives ac and ca, and the third gives bc and cb. Thus the rectangular cross parts are 2ab, 2ac, and 2bc when paired by type, but physically they are six separate rectangles: two ab, two ac, and two bc. Therefore (a+b+c)^2=a²+b²+c²+2ab+2ac+2bc. A, B, and C undercount these pairwise rectangles. Memory cue: with three side parts, choose three pairs, and each pair appears twice: 3×2=6.

Answer: A, (x + 4)(x - 3). In a rectangle model, multiplying the two side expressions gives the total area. Expand the first option step by step: (x + 4)(x - 3) = x·x - 3x + 4x - 12 = x² + x - 12. Thus, the x² part comes from x by x, the two cross-rectangles give -3x and +4x, and the corner gives -12. Option B gives x² - 7x + 12, so both the middle term and constant are wrong. Option C gives x² + 4x - 12, so its x-term is incorrect. Option D gives x² + 11x - 12, also incorrect. Memory cue: for (x + p)(x + q), check p + q for the x-coefficient and pq for the constant term; here 4 + (-3) = 1 and 4(-3) = -12.

Answer: A, 4 and 10. To split the quadratic into a rectangle model, we need two numbers whose sum is the coefficient of x and whose product is the constant term. The required sum is 14 and the required product is 40. Since 4 + 10 = 14 and 4×10 = 40, the expression factors as x² + 14x + 40 = (x + 4)(x + 10). Option B has the correct-looking sum, 5 + 9 = 14, but its product is 45, not 40. Option C has product 40, but its sum is 22, not 14. Option D has sum 14, but its product is 48. Thus both tests, not just one, are necessary. Memory cue: in a monic quadratic x² + bx + c, search for numbers adding to b and multiplying to c.

Answer: B, 4x. Multiply the dimensions using the area model: (x - 5)(x + 9) gives four parts, x·x = x², x·9 = 9x, (-5)·x = -5x, and (-5)·9 = -45. The question asks only for the two rectangles containing x, so combine the cross terms: 9x + (-5x) = 4x. Therefore the required total is 4x. Option A, 14x, adds the magnitudes 9 and 5 and ignores the negative sign. Option C has the wrong sign because 9x is larger than 5x. Option D treats the constant corner -45 as though it were an x-rectangle, but it contains no x. Memory cue: in a signed rectangle, preserve the sign of every side before combining like terms.

Answer: A. The area of a square is side × side, or side². Thus a square with side x + 4 has area (x + 4)², and a second square with side x - 4 has area (x - 4)². Subtracting the second area from the first gives exactly (x + 4)² - (x - 4)². Option B describes only one square-like area and cannot represent a difference of two square areas. Option C gives x² - x² = 0 because the squares are equal, not the given expression. Option D involves a circle, whose area is not represented by the square expressions here. The same model also connects to the identity A² - B² = (A + B)(A - B), but the visual meaning comes first: two square regions with different side lengths.

Answer: B, 16x. Apply the difference-of-squares identity A² - B² = (A + B)(A - B), taking A = x + 4 and B = x - 4. First calculate A + B = (x + 4) + (x - 4) = 2x. Next calculate A - B = (x + 4) - (x - 4) = 8; the second subtraction changes the sign of -4. Hence the product is 2x×8 = 16x. Option A forgets that the first factor is 2x rather than x. Options C and D incorrectly treat the expression as if it were a simple square expansion; they do not use the difference-of-squares structure. Memory cue: calculate the sum and difference separately, especially when the second bracket contains a minus sign.

Answer: B, 2ax. Imagine the large square of side x + a divided into four parts by separating each side into lengths x and a. The x-by-x region has area x², and the a-by-a corner square has area a², as stated. The remaining two rectangles each have dimensions x and a, so each has area ax. Adding those two equal rectangles gives ax + ax = 2ax. Option A is only the area of one rectangle, not both. Option C adds the two square areas and leaves out the rectangles. Option D includes an extra factor of a and has the wrong dimensions. This is the middle term in (x + a)² = x² + 2ax + a². Memory cue: the two identical cross-rectangles create the doubled product 2ax.

Answer: C, the middle term 12xy. Use (A + B)² = A² + 2AB + B² and (A - B)² = A² - 2AB + B², with A = 2x and B = 3y. In both expressions, A² = 4x² and B² = 9y² remain positive and unchanged. The cross contribution is 2AB = 2(2x)(3y) = 12xy. It is +12xy in the first square and -12xy in the second, so only this middle part changes sign. Option A is wrong because (2x)² stays 4x². Option B is wrong because (3y)² stays 9y². Option D incorrectly says both square terms change. Memory cue: changing plus to minus affects the cross term, not the two pure square terms.

Answer: A, (6x + 1)(6x - 1). Recognise 36x² as (6x)² and 1 as 1². Therefore the expression is (6x)² - 1². Applying A² - B² = (A + B)(A - B) gives (6x + 1)(6x - 1). Option B uses 36x as the side instead of its square root, so its product begins with 1296x². Option C has unequal and unsuitable factors; its x² coefficient is 6, not 36, and its constant is -1. Option D uses 18x, whose square would be 324x², so it is also too large. The visual interpretation is two squares with sides 6x and 1; their difference can be rearranged into the stated rectangle. Memory cue: before factoring a difference of squares, take the square root of each square term, not the term itself.

Correct answer: A. The outer figure is a square because both of its sides have length a+b. Divide each side into parts a and b. The four regions are an a by a square with area a², an a by b rectangle with area ab, a b by a rectangle with area ab, and a b by b square with area b². Adding them gives a²+ab+ab+b²=a²+2ab+b²=(a+b)². Option B describes dimensions associated with a difference of squares and omits the two ab regions. Option C uses the wrong outer side, a−b, and therefore cannot model (a+b)². Option D separates the two small squares and leaves out the connecting rectangles, so it misses the 2ab part. Memory cue: a binomial square always has two equal cross-rectangles.

Correct answer: B. Expand the square: (x+k)²=x²+2kx+k². Comparing this with the given information, the middle term 2kx is 18x, so 2k=18 and k=9. The corner term gives a second check: k²=81, and the positive side value consistent with the plus sign is k=9. Option A copies the middle coefficient without dividing by 2. Option C uses the corner area itself as the side; since k²=81, the side is its square root, not 81. Option D is not supported by either condition and results from an incorrect halving or root selection. A useful check is to substitute k=9: (x+9)²=x²+18x+81, exactly matching the model.

The direct answer is B: the length is x + y. The area of the rearranged rectangle is still x² − y² because rearranging pieces does not change total area. Use the difference-of-squares identity: x² − y² = (x − y)(x + y). One factor is already given as the height, x − y, so the remaining factor must be the length, x + y. A repeats the height and would give area (x−y)², not x²−y². C would give x²y − xy², which is a different expression and has three factors/variables in an unsuitable form. D is twice the height and would give 2(x−y)², again not the required area. Memory cue: for a²−b², write ‘difference times sum’: (a−b)(a+b).

The direct answer is D: the unshaded part is a². The whole square has side a+b, so its area is (a+b)²=a²+2ab+b². The shaded region is given as 2ab+b². Subtract it from the whole area: (a²+2ab+b²)−(2ab+b²)=a². This also matches the picture idea: the unshaded corner is a square whose side is a, so its area is a². A, 2ab, represents the two rectangular strips together, not the remaining square. B, ab, represents only one rectangle. C includes b², which is already part of the shaded region and therefore cannot be the unshaded remainder. Always remember that shaded area plus unshaded area must equal the complete square area. This check makes the answer unambiguous.

The direct answer is D. Start with an x-by-x square, whose area is x². Remove a smaller 8-by-8 square, whose area is 64. The remaining area is therefore x²−64. By cutting and rearranging the remaining pieces, they form a rectangle whose sides are x−8 and x+8; hence the area is (x−8)(x+8). Option A adds two rectangles and would represent extra positive area rather than subtracting 64. Option B forms (x+8)², which contains x²+16x+64, not x²−64. Option C forms (x−8)², which contains x²−16x+64, also different. D correctly shows a difference of two squares: x²−8²=(x−8)(x+8). Memory cue: a large square minus a small square becomes a rectangle with ‘difference’ and ‘sum’ as its sides.

The direct answer is B: 4pq. The border is the area of the outer square minus the area of the inner square. The outer area is (p+q)² and the inner area is (p−q)², so border area = (p+q)²−(p−q)². Apply a²−b²=(a−b)(a+b): [(p+q)−(p−q)] [(p+q)+(p−q)] = (2q)(2p)=4pq. Option A is only half of the complete result. Option C, p²−q², is the product (p−q)(p+q), which is related to the identity but is not the difference between these two square areas. Option D comes from adding rather than subtracting the square areas and is therefore wrong. A safe visual check is that the border must equal outer area minus inner area, not their sum.