Visual models of identities

Direct answer: A is correct. Start with a large square of side \(a\), so its area is \(a^2\). To make a square of side \(a-b\), remove a strip of width \(b\) from one side; its area is \(ab\). Remove another strip of width \(b\) from the adjacent side; its area is another \(ab\). The small corner where the strips overlap has dimensions \(b\) by \(b\), so its area is \(b^2\). That corner has been included in both removed strips and therefore subtracted twice. We add it back once, giving \(a^2-ab-ab+b^2=a^2-2ab+b^2\). B is wrong because the corner is removed as part of both strips. C is wrong because the original \(a^2\) is not increased. D is wrong because \(2ab\) is subtracted, not added. Memory cue: in a square of a difference, subtract two \(ab\) parts and restore the doubly counted \(b^2\) corner.

Direct answer: A is correct. In a rectangle model, split one side into lengths \(x\) and \(2\), and the other side into \(x\) and \(7\). Each small region is found by multiplying its horizontal and vertical side lengths. The four products are \(x\cdot x=x^2\), \(2\cdot x=2x\), \(7\cdot x=7x\), and \(2\cdot7=14\). Their sum is \(x^2+2x+7x+14=x^2+9x+14\), which agrees with the distributive expansion. B is wrong because \(9x\) is the combined linear contribution, not one of the four original regions, and \(2\) is not the constant product. C is wrong because 9 is a sum, not the product of the constant sides. D is wrong because \(14x\) is not produced by any pair of side lengths. The safe method is to multiply every part of one bracket by every part of the other bracket exactly once.

Direct answer: C, \(9x\), is correct. Draw a four-part area model for \((2x+1)(x+4)\). The product \(2x\cdot x\) gives the quadratic part \(2x^2\), while \(1\cdot4\) gives the constant part 4. The two cross-products create the linear terms: \(2x\cdot4=8x\) and \(1\cdot x=x\). Since both are like terms, combine them: \(8x+x=9x\). Thus the full expansion is \(2x^2+9x+4\). A is wrong because 5x does not result from either correct sum. B is only the first cross-product and ignores the additional \(x\). D is only the second constant-times-variable product and ignores \(8x\). Memory cue: the total linear term in a two-binomial product comes from both cross-products, not just one.

Direct answer: A is correct. The area of a rectangle equals length multiplied by breadth, so calculate \((3y+2)(y+5)\). Multiply each term in the first bracket by each term in the second: \(3y\cdot y=3y^2\), \(3y\cdot5=15y\), \(2\cdot y=2y\), and \(2\cdot5=10\). Now combine like terms: \(15y+2y=17y\). Therefore the total area is \(3y^2+17y+10\). B is wrong because it omits the \(2y\) contribution. C is wrong because it treats the linear terms as if they were subtracted or incorrectly combined. D is wrong because it places the constant 17 and the linear coefficient 10 in the wrong roles. The visual model is useful because its four regions ensure that no multiplication pair is forgotten.

Direct answer: D, 6, is correct. In the grid for \((4p+3)(p+2)\), separate the variable parts \(4p,p\) from the constants 3 and 2. The constant corner is formed by multiplying the two constant side lengths, not by adding them. Thus its area is \(3 ext{ × }2=6\). The other grid regions are \(4p ext{ × }p=4p^2\), \(4p ext{ × }2=8p\), and \(3 ext{ × }p=3p\), so the full expansion would be \(4p^2+11p+6\). A and B are merely the individual constants, not their product. C is their sum, \(3+2\), which does not represent the corner area. The memory cue is: a corner in a product grid is always obtained by multiplying the two entries meeting at that corner.

Direct answer: A is correct. Add the areas of all four regions in the square model: one region contributes \(a^2\), one contributes \(b^2\), and the two equal rectangles contribute \(ab+ab=2ab\). The total is therefore \(a^2+2ab+b^2\). By the square-of-a-sum identity, this equals \((a+b)^2\). B is wrong because a square of a difference has negative rectangular contributions, \(a^2-2ab+b^2\). C is incomplete because it leaves out both \(ab\) rectangles. D represents a difference of squares, which has a different structure and no middle \(2ab\) term. A useful memory cue is that two positive cross-rectangles mean plus \(2ab\); two negative cross-rectangles mean minus \(2ab\).

Direct answer: B, \(12x\), is correct. Begin with a large square of side \(x\), whose area is \(x^2\). To reduce both dimensions from \(x\) to \(x-6\), remove one strip of width 6 and length \(x\) from one side; its area is \(6x\). Remove an equal strip from the adjacent side; its area is another \(6x\). The total area of the two strips, counted as the two strip regions before correcting their overlap, is \(6x+6x=12x\). The small overlapping corner has area \(6 ext{ × }6=36\), and it is handled separately when writing \((x-6)^2=x^2-12x+36\). A counts one strip only. C uses an incorrect product. D is the corner square, not the total of the strips. Memory cue: two strips of width \(a\) along an \(x\)-square contribute \(2ax\).

Direct answer: A is correct. The area of a square is the square of its side, so the expression is \((2x-5)^2\). Use \((a-b)^2=a^2-2ab+b^2\), with \(a=2x\) and \(b=5\). Then \(a^2=(2x)^2=4x^2\), \(-2ab=-2(2x)(5)=-20x\), and \(b^2=25\). Hence the expansion is \(4x^2-20x+25\). B incorrectly uses \(-10x\), as though the factor 2 in \(2x\) were missed. C squares \(2x\) incorrectly. D has the wrong sign; the middle term is negative because the binomial contains subtraction. Options A, B, and C have the same general term pattern but incorrect calculations in the latter two, while D has a sign error.

Direct answer: B, \(32x\), is correct. Expand the first square: \((x+8)^2=x^2+16x+64\). Expand the second: \((x-8)^2=x^2-16x+64\). Now subtract the entire second expression: \((x^2+16x+64)-(x^2-16x+64)\). The \(x^2\) terms cancel, and the constant 64 terms also cancel. The remaining terms are \(16x+16x=32x\). A is only one of the two linear contributions. C is not the result of this subtraction and leaves an incorrect combination of terms. D incorrectly treats the constants as if they produced a factor of x. A faster check uses the difference-of-squares identity: \(u^2-v^2=(u-v)(u+v)\), giving \((16)(2x)=32x\).

Direct answer: A is correct. A square with side \(a+b+c\) has area \((a+b+c)^2\). In the visual partition there are three square regions, with areas \(a^2,b^2,c^2\). For each pair of different lengths, there are two rectangular regions: the pair \(a,b\) gives \(2ab\), the pair \(b,c\) gives \(2bc\), and the pair \(c,a\) gives \(2ca\). Thus \((a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\), so all three pair terms in A must appear. B is incomplete because it includes only one copy of each pair. C omits every rectangular region. D has three factors and is not produced by multiplying two side lengths in an area model. Memory cue: in the square of a three-term sum, every pair appears twice.

The correct answer is A: the areas are a², b², and c². In a visual model, the side of the large square is divided into three lengths, a, b, and c. The small square belonging to length a has both its sides equal to a, so its area is a×a=a². Similarly, the other two small squares have areas b×b=b² and c×c=c². The products ab, bc, and ca describe rectangular pieces because their side lengths are different. The expressions 2ab, 2bc, and 2ca represent the combined areas of pairs of equal rectangles in the expansion of (a+b+c)², not the individual small squares. Option D mixes unrelated quantities and does not describe the three square regions. Memory cue: equal side × equal side gives a square term; two different side lengths give a rectangle term.

The correct answer is B: 4mn. The two areas are (m+n)² and (m−n)². Expand them carefully: (m+n)²=m²+2mn+n², while (m−n)²=m²−2mn+n². Now subtract the second area from the first: [m²+2mn+n²]−[m²−2mn+n²]. The m² terms cancel and the n² terms cancel. The remaining expression is 2mn−(−2mn)=2mn+2mn=4mn. Therefore B is correct. A is only one middle term, not the difference between the two complete areas. C is the difference of m² and n², which belongs to another identity. D incorrectly doubles the square terms even though they cancel. Memory cue: subtracting the two square expansions makes the equal square terms disappear and doubles the opposite middle terms.

The correct answer is A: x². In the square model of (x+3)², one region is formed by multiplying x by x, so that region has area x². In the rectangle model of (x+3)(x+4), the x part of the first side and the x part of the second side also form an x-by-x region, again with area x². The expansions confirm this: (x+3)²=x²+6x+9, whereas (x+3)(x+4)=x²+7x+12. Thus x² appears in both models. Option B, 12, is the constant term only in the rectangle model. Option C, 7x, is the combined linear part of the rectangle model and is not the same as the 6x part of the square model. Option D, 9, is the corner square in the first model but not in the second. The common visual block is the x-by-x square.

The correct answer is B. Since both sides are x+5, the rectangle is actually a square with area (x+5)². Use the square identity (p+q)²=p²+2pq+q², with p=x and q=5. This gives x²+2(x)(5)+5²=x²+10x+25. The two side strips in the visual model each have area 5x, so together they form the middle term 10x. Option A is false because the strips have equal dimensions and equal areas. Option C is false because the constant corner is 5×5=25, not 10. Option D is false because the central x-by-x region contributes x². The important distinction is that 10x comes from two rectangles, while 25 comes from the small square. Memory cue: in (x+k)², the middle term is 2kx and the constant term is k².

The correct answer is D: 2ab-b². Start with the square of side a, whose area is a². A square of side a-b has area (a-b)²=a²-2ab+b². Therefore, the amount missing from this smaller square to reach the full a² square is a²-(a²-2ab+b²)=2ab-b². Equivalently, the model contains two strips of area ab each, but their corner overlap of area b² must be counted only once, so the added region is ab+ab-b²=2ab-b². Option A includes only the overlap and is incomplete. Option B counts only one strip. Option C is not an area expression with the required structure and omits b. Memory cue: two strips contribute 2ab, then subtract their common b² overlap.

The direct answer is B. When the side of the whole square is split into lengths a and b, the complete side is a+b. Therefore the whole area is (a+b)². By the square identity, (a+b)² = a² + 2ab + b². The visual partition has one a by a square with area a², one b by b square with area b², and two rectangles, each with area ab. Adding all four non-overlapping pieces gives a² + ab + ab + b² = a²+2ab+b². Option A omits the two rectangular parts. Option B includes every part exactly once. Option C represents subtraction and would require a difference of side lengths. Option D doubles the square areas and omits the rectangles. Memory cue: two mixed rectangles always create 2ab in the addition-square model.

The direct answer is B, 2xy. In the square model, removing the corner square of side y leaves two congruent rectangular strips. Each rectangle has one dimension x and the other dimension y, so the area of each is x × y = xy. Since there are two equal rectangles, their combined area is xy + xy = 2xy. Option A gives the area of only one rectangle, not both. Option B correctly counts the two rectangles. Option C adds the areas of two squares and does not describe the rectangular strips. Option D is a length expression, not an area expression; it also does not multiply the dimensions. Memory cue: count identical regions and multiply the area of one region by their number.

The direct answer is D. The area of the rectangle is length × breadth = (a+b)(a−b). Apply the identity (u+v)(u−v)=u²−v² with u=a and v=b. The result is a²−b². Option A is the expansion of (a+b)², which would require both sides to be a+b. Option B is the expansion of (a−b)², which would require both sides to be a−b. Option C leaves out the interaction of the two factors and is not generally equal to their product. Option D is correct because the positive and negative mixed terms cancel: a²−ab+ab−b² = a²−b². Memory cue: opposite signs in matching brackets cause the middle terms to cancel.

The direct answer is A. Begin with the large square of area m². Each strip has area mn, and two strips are removed, so subtract 2mn. The corner square of area n² is included in both strips and was therefore subtracted twice; adding it back once corrects the double subtraction. The result is m²−2mn+n², which equals (m−n)². Option A follows the exact remove-and-restore process. Option B adds the strips and describes enlargement by n, not subtraction. Option C is the difference-of-squares identity and has no +n² correction term. Option D is false in general because (m+n)² also contains 2mn. Memory cue: subtract two strips, then add the overlapped corner once.

The direct answer is B: r²+2rs+s². The model contains one central square with area r², two separate strips with area rs each, and one corner square with area s². Because these pieces together form the whole region without overlap, add them once: r² + rs + rs + s² = r² + 2rs + s². This is also the expansion of (r+s)². Option A counts only one of the two strips, so it misses rs. Option B includes the central square, both strips, and the corner square exactly once. Option C subtracts s² even though it is part of the total region. Option D doubles r² and leaves out both strips. Memory cue: two equal mixed areas rs combine to 2rs.

The correct answer is D: add d². Begin with a square of area c². Removing one strip of area cd gives c²−cd. Removing the second strip gives c²−2cd. But the two strips overlap in a small corner square of side d, whose area is d². That corner has been removed twice, although it should be removed only once. To correct this double subtraction, add d² once. Thus the result is c²−2cd+d², which is (c−d)². Option A is wrong because d² is not subtracted at the correction stage; it must be restored. Option B restores only cd, not the doubly counted corner. Option C subtracts 2d² and makes the area even smaller. Memory cue: in a difference-of-squares picture, the overlap removed twice is added back once.

The correct answer is C: z−5 and z+5. The large square has area z². The removed square has area 5²=25. Therefore, the remaining area is z²−25. This is a difference of squares: z²−5²=(z−5)(z+5). Hence the same area can be represented by a rectangle whose sides are z−5 and z+5. Option A gives area 5z, which is not generally z²−25. Option B gives (z−5)² = z²−10z+25, a different expression. Option D gives (z+5)² = z²+10z+25, also different. The factors are valid as lengths when z>5, as required for the smaller square to fit inside the larger one. Memory cue: a²−b² always factors into (a−b)(a+b), not two equal factors.

The correct answer is A: x²+9x+14. Split the first side into x and 2, and the second side into x and 7. The rectangle then has four parts: x×x=x², x×7=7x, 2×x=2x, and 2×7=14. Add them: x²+7x+2x+14 = x²+9x+14. This is also the distributive expansion (x+2)(x+7). Option B places 14 as the coefficient of x and 9 as the constant, reversing the roles of the cross-products and corner. Option C uses 5x, which would come from adding 2 and 3, not 2 and 7. Option D has the wrong sign for the constant; both dimensions contain positive numbers, so the constant product is +14. Memory cue: in (x+a)(x+b), the middle coefficient is a+b and the constant is ab.

The correct answer is B: t²−t−12. The area of a rectangle is the product of its side lengths, so write (t+3)(t−4). Multiply every term in the first bracket by every term in the second: t×t=t², t×(−4)=−4t, 3×t=3t, and 3×(−4)=−12. Combining like terms gives t²−4t+3t−12 = t²−t−12. Option A incorrectly adds the middle contributions as 4t+3t instead of respecting the negative sign. Option C changes the signs of the middle and constant terms. Option D has the wrong constant sign and the wrong net coefficient of t. A tile model makes the four products visible, which helps prevent missed or incorrectly signed terms. Memory cue: use FOIL or four products, then combine like terms only at the end.

The correct answer is C: k+6. The small square has area 36, so its side is √36=6. In a square-completion model, the total strip area around the k² square is 2×k×6=12k, exactly as given. Therefore the whole square has side k+6, and its area is (k+6)² = k²+12k+36. Option A would produce a strip term 24k and a small square of area 144, so it does not fit. Option B gives a negative-side form and would produce −12k in the expansion. Option D gives leading term 36k² when squared, not k². The key is to find the side of the small 36-square first, then use the strip coefficient as a check. Memory cue: 12k=2×k×6, so the added side is 6.