Visual models of identities

The direct answer is D: a=6 and b=7, in either order. Expand the rectangle model: (x+a)(x+b)=x²+(a+b)x+ab. The constant corner requires ab=42, while the two middle strips are ax and bx, whose sum is (a+b)x=13x. So we need two numbers with product 42 and sum 13. The factor pairs of 42 are 1 and 42, 2 and 21, 3 and 14, and 6 and 7. Only 6+7=13, so D satisfies both conditions. A has the correct product but sum 43. B has product 42 but sum 17. C has product 42 but sum 23. The key lesson is to use both the corner tile and the middle strips; matching only the product can lead to a wrong choice. Since multiplication is commutative, the order of 6 and 7 does not matter.

The direct answer is B: 4x²−9. The rectangle area is the product of its side lengths: (2x+3)(2x−3). This has the form (a+b)(a−b), where a=2x and b=3. By the difference-of-squares identity, (a+b)(a−b)=a²−b². Therefore the area is (2x)²−3²=4x²−9. Option A adds the squares instead of subtracting them. Option C uses an incorrect coefficient for the leading term. Option D is the expansion of neither the given product nor the square of a relevant binomial; it introduces an unnecessary −6x term. The opposite signs in the two side lengths make the cross-products cancel: −6x+6x=0. Memory cue: conjugate factors, such as (a+b) and (a−b), always produce a²−b² with no middle term.

The direct answer is A. Divide each side of the large square into two parts, p and q. This creates four smaller regions: a p by p square with area p², a q by q square with area q², and two rectangles, each measuring p by q and therefore having area pq. Adding all four regions gives p² + pq + pq + q² = p² + 2pq + q². This is exactly the identity (p+q)². B is wrong because one pq rectangle is missing. C has the wrong shape and area, while D counts three squares and does not represent the two cross-products. Memory cue: the middle term 2pq comes from two equal pq rectangles.

The direct answer is D. Compare the expression with the identity (u−v)²=u²−2uv+v². Here 49a²=(7a)² and 25b²=(5b)². The middle term produced by (7a−5b)² is −2(7a)(5b)=−70ab, exactly the given middle term. Thus 49a²−70ab+25b²=(7a−5b)², represented by a square whose side is 7a−5b. A uses the full terms as a side and would create much larger powers. B gives a positive middle term +70ab. C represents a difference of squares, which would produce 49a²−25b² and no −70ab term. Memory cue: a negative middle term means the two parts of the square side have opposite signs.

The direct answer is D. Make a rectangle whose side lengths are x+7 and x+8, then split each side into x and its constant part. The four cells are x by x, x by 8, 7 by x, and 7 by 8. Their areas are x², 8x, 7x, and 56. The two middle strips are therefore 7x and 8x; together they form the middle term 15x. Hence (x+7)(x+8)=x²+15x+56, so m=15. A confuses a corner with a strip, B gives a sum rather than the two separate strips, and C lists only side lengths rather than areas. Memory cue: each x meets each constant once, producing the two x-strips.

The direct answer is C: four strips. Expand or use the difference-of-squares identity: (x+3)²−(x−3)²=[(x+3)−(x−3)][(x+3)+(x−3)]=(6)(2x)=12x. Each stated strip has area 3x, so the number of strips is 12x÷3x=4, for x≠0. The same result follows by expansion: x²+6x+9 minus (x²−6x+9) leaves 12x. A counts only half the difference, B does not match 12x, and D would represent 18x, too large. In the picture, the symmetric square models cancel their x² and 9 corners, leaving four strips of size 3x. Memory cue: 12x is four copies of 3x.

The direct answer is B: 16x. The border is the outer square minus the empty inner square. Its area is (x+4)²−(x−4)². Use U²−V²=(U−V)(U+V), with U=x+4 and V=x−4. Then U−V=8 and U+V=2x, so the border area is 8·2x=16x. A is only half this result. C is an unsimplified product-related expression and does not equal the border area; in fact x²−16 is the product (x+4)(x−4). D is a constant and cannot represent the changing border area. For actual positive side lengths, one normally needs x>4. Memory cue: outer square minus inner square, then use difference of squares.

The direct answer is A. Multiply each row label by each column label to obtain the four signed cells. The products are 3x·x=3x², 3x·2=6x, (−5)·x=−5x, and (−5)·2=−10. Add them with their signs: 3x²+6x−5x−10=3x²+x−10. B incorrectly adds the middle terms as 6x+5x instead of recognising that −5x is negative. C pairs the labels incorrectly and does not represent the grid cells. D omits both middle cells, even though they are essential to the total. In a signed area model, negative tiles are algebraic contributions; they are not simply ignored. Memory cue: fill every cell first, attach its sign, then combine like terms.

The correct answer is A: x^2+xy+y^2. Begin with the identity (x+y)^2=x^2+2xy+y^2. In an area model, the large square has area x^2, the small square has area y^2, and there are two rectangular strips. Each strip has length x and width y, so each has area xy. Therefore the two strips together contribute xy+xy=2xy. If the model mistakenly includes only one strip, one xy is missing, and the displayed total becomes x^2+xy+y^2. Option A is correct because it shows exactly one middle rectangle. Option B, x^2+2xy, leaves out the y^2 corner square. Option C, x^2+y^2, leaves out both xy rectangles, not just one. Option D, 2xy+y^2, leaves out the main x^2 square. Memory cue: the middle term 2xy means two equal xy rectangles; never count it as one rectangle.

The correct answer is A: (a+b)^2=a^2+2ab+b^2. Start with one square of side a, whose area is a^2. Add a rectangular strip of dimensions a by b along one side; its area is ab. Add a second strip of the same dimensions along the adjacent side; its area is another ab. The remaining corner is a b by b square with area b^2. The completed figure has side a+b, so its area is (a+b)^2, equal to a^2+ab+ab+b^2=a^2+2ab+b^2. Option A describes addition of both strips and the corner. Option B represents a difference-of-squares arrangement, where a region is removed rather than added. Option C also involves removing strips to make a smaller square of side a-b. Option D is an equivalent difference-of-squares identity but uses a rectangle with sides a+b and a-b, not two added strips to an a-square. Memory cue: added strips give plus signs; removed strips give minus signs.

The correct answer is A: (5x-1) and (5x+1). Recognize 25x^2 as (5x)^2 and 1 as 1^2. Thus the expression is a difference of two squares: (5x)^2-1^2. Using p^2-q^2=(p-q)(p+q), with p=5x and q=1, we obtain (5x-1)(5x+1). In the rearranged visual model, these two factors become the rectangle's side lengths. Option A is correct because multiplying the sides gives 25x^2-1. Option B would multiply to 625x^2-1, so it uses 25x instead of its square root, 5x. Option C gives (5x-1)(x+1), whose leading term is 5x^2 and therefore cannot match 25x^2. Option D is a square of (5x+1), producing positive middle and constant terms, not a difference. Memory cue: for U^2-V^2, take the difference and sum of the square roots.

The correct answer is A: 7xy. The expression x^2+14xy+49y^2 is the expansion of (x+7y)^2. In the square model, x^2 is one corner square and 49y^2=(7y)^2 is the other corner square. The middle term comes from two equal rectangles, each having dimensions x and 7y. Therefore the area of each rectangle is x times 7y, or 7xy. Together they contribute 7xy+7xy=14xy. Option A gives the area of one rectangle. Option B is the combined area of both middle rectangles, not the area of one. Option C is the area of the (7y)-by-(7y) corner square. Option D is a side length, not an area, and also lacks square units. A useful check is to identify the coefficient of the middle term: in (p+q)^2, the 2pq term is split into two equal pq regions.

The correct answer is B: 8ab. The required area is the difference between the two square areas: (2a+b)^2-(2a-b)^2. Use the identity (p+q)^2-(p-q)^2=4pq. Here p=2a and q=b, so the difference is 4(2a)(b)=8ab. You can also expand: (4a^2+4ab+b^2)-(4a^2-4ab+b^2)=8ab. Visually, the larger square contains the smaller square plus the remaining strips; together those strips have area 8ab. Option A, 4ab, results from forgetting that p is 2a rather than a, so it is half the required value. Option C is the difference of the two side quantities squared in another order and is not the requested result. Option D is only part of an expansion and omits the full cancellation process. Memory cue: (p+q)^2-(p-q)^2 always equals 4pq; substitute p=2a carefully.

The correct answer is A: 8x. The remaining outer area is the area of the large square minus the area of the smaller square: (x+2)^2-(x-2)^2. Apply the difference-of-squares identity with P=x+2 and Q=x-2. The result is [(x+2)-(x-2)] [(x+2)+(x-2)] = 4×2x=8x. Visually, the leftover region consists of two strips around the smaller square. Each strip has width 4, while their combined length contribution gives the same total area 8x; the identity is the safest calculation. Option A is correct. Option B, 4x, describes only half of the total strip area in this comparison. Option C is the product (x+2)(x-2)=x^2-4, not the difference of the square areas. Option D is only a constant and ignores the variable length x. Memory cue: when one square is inside another, subtract their areas, then use difference of squares rather than expanding unnecessarily.

The direct answer is A: (3x + 2)(5x − 4). In a rectangle model, multiply every part of one side by every part of the other side. The four areas are (3x)(5x) = 15x², (3x)(−4) = −12x, 2(5x) = 10x, and 2(−4) = −8. Adding them gives 15x² − 12x + 10x − 8, which simplifies to 15x² − 2x − 8. Thus the two middle rectangles really do have areas −12x and 10x. Option B expands to 15x² + 2x − 8, so its middle term has the wrong sign. Option C gives a different leading structure and does not produce the required four areas. Option D has middle terms −20x and 6x, whose sum is −14x, not −2x. Remember that signed rectangle areas must be added with their signs; do not combine the two cross terms by magnitude alone.

The direct answer is A: 4c(a + b). Begin by treating the repeated group a + b as one quantity, say u. The expression becomes (u + c)² − (u − c)². Using the difference-of-squares identity, P² − Q² = (P − Q)(P + Q), we get [(u+c) − (u−c)] [(u+c) + (u−c)] = (2c)(2u) = 4uc. Replacing u by a+b gives 4c(a+b). Option B is only half of the correct result because both the length 2c and the length 2(a+b) contribute. Option C incorrectly multiplies a, b, and c separately; the expression contains the grouped sum a+b, not ab. Option D keeps square terms that cancel during subtraction, so it cannot represent the remaining strip. A useful cue is: (u+c)² − (u−c)² always leaves four times u c.

The direct answer is A: (a+b)² = a² + 2ab + b². A square with side a+b can be divided into one a-by-a square, one b-by-b square, and two rectangles, each with area ab. Their total area is a² + ab + ab + b² = a² + 2ab + b². If the a and b strips are interchanged, the pieces may move or change position, but their dimensions and total area do not change. Therefore the same identity remains true. Option B is a different identity involving the difference a−b; it is not the area of the stated square. Option C is false because a²−b² is (a−b)(a+b), not generally (a−b)². Option D forgets the two rectangular areas 2ab. Memory cue: rearranging pieces preserves total area, and the two cross-rectangles create the middle term 2ab.

The governing concept is the perfect-square identity (x + y)² = x² + 2xy + y². Compare the given trinomial with this form. Since 81 = 9² and 18m = 2×m×9, we can rewrite m² + 18m + 81 as m² + 2(m)(9) + 9², which equals (m + 9)². If the area of a square is (m + 9)², its side is m + 9, so option D is correct. Option A would expand to m² + 12m + 36, option B would produce m² − 18m + 81 because its middle term is negative, and option C would give m² + 36m + 324. Thus only D reproduces the original square exactly. The visual model represents the two-variable side components and the constant 9, making the identity a geometric as well as algebraic interpretation.

The direct answer is B: 11r and −11r. Multiply the two conjugate expressions using a rectangle model: (r+11)(r−11). The four signed regions are r², −11r, 11r, and −121, depending on which side parts are paired. The two linear regions have equal magnitude, 11r, but opposite signs, so they cancel: 11r + (−11r) = 0. The remaining area is r² − 121, which is the difference-of-squares identity. Option A names two positive regions, so they would add rather than cancel. Option C pairs the square term with the constant term; those are not like terms and are not the visual cancellation being asked about. Option D contains opposite constant regions, but the model has only one constant product, −121, not a positive and negative pair. Remember: cancellation requires equal size and opposite signs.

The direct answer is C: 9x² − 16. The product is of the form (a+b)(a−b), whose identity is a²−b². Here a=3x and b=4. Therefore the area is (3x)²−4² = 9x²−16. In the rectangle picture, the two cross-regions are equal and opposite, so the 12x terms cancel; only the large square 9x² and the negative corner 16 remain. Option A incorrectly adds the two square areas, as if the expression were a sum of squares. Option B squares 3x incorrectly and should have 9x², not 3x². Option D is the expansion of (3x−4)², which is a different expression and includes unwanted linear terms. A useful cue is that conjugate factors have opposite signs and always produce a difference of squares.

The direct answer is D: 6 and 9. Suppose the rectangle sides are (x+a) and (x+b). Multiplication gives x²+(a+b)x+ab. To match x²+15x+54, the constants must satisfy a+b=15 and ab=54. The pair 6 and 9 satisfies both conditions: 6+9=15 and 6×9=54. Thus the rectangle is (x+6)(x+9). Option A has product 54 but sum 29, so its middle coefficient is wrong. Option B has sum 15 but product 50, so its constant term is wrong. Option C has product 54 but sum 21, again giving the wrong middle term. Option D is the only pair satisfying both requirements simultaneously. In a tile model, always check the sum for the coefficient of x and the product for the constant tile; checking only one condition can lead to a tempting but incorrect choice.

Correct answer: B, 2 rectangles. Imagine that each side of the large square is divided into lengths a, b, and c. In the area model, the horizontal part of length a and the vertical part of length b form one rectangle with area ab. The horizontal part of length b and the vertical part of length a form another rectangle with area ba. Since multiplication is commutative, ba equals ab. Thus the two regions together contribute ab+ab=2ab. Option A counts only one orientation and misses the second rectangle. Option C incorrectly counts three rectangles, although there are only two ways to pair a and b. Option D counts too many regions and has no basis in the model. Memory cue: in (a+b+c)^2, every product of two different terms appears twice: 2ab, 2bc, and 2ca.

Correct answer: C, 2uv+2vw+2wu. Divide each side of the large square into lengths u, v, and w. The three same-term regions are squares with areas u^2, v^2, and w^2; these are not mixed rectangles. For the pair u and v, the model contains two rectangles, with areas uv and vu, together giving 2uv. Similarly, the pair v and w gives 2vw, and the pair w and u gives 2wu. Adding all mixed regions gives 2uv+2vw+2wu. Option A counts each pair only once and therefore gives half the mixed area. Option B contains only the three square regions. Option D has three factors and does not represent a rectangle area from this square model. Remember: every unlike pair occurs in two orientations.

Correct answer: D, (x+8) and (x+15). In a rectangle area model, the x^2 part comes from x multiplied by x. The two linear strips reveal the remaining pairings: x multiplied by 8 gives 8x, and x multiplied by 15 gives 15x. The corner formed by the constant lengths must then have area 8×15=120, which matches the given constant part. Therefore the complete product is (x+8)(x+15)=x^2+15x+8x+120=x^2+23x+120. Option A would produce x^2+29x+120, so its linear parts do not match. Option B does not produce the listed strips or corner. Option C gives 10x and 12x, not 8x and 15x. A useful check is to multiply the two proposed side constants and compare with the corner area.

Correct answer: A, add one more 6x strip. The square has side 2x+3, so its area is (2x+3)^2. In the visual model, the 2x by 3 parts occur twice: one rectangle has area (2x)(3)=6x, and the other has area (3)(2x)=6x. Together they form the middle term 6x+6x=12x. The 4x^2 region is correct because it comes from (2x)(2x), and the 9 corner is also correct because it comes from 3×3. Option B would remove a required region and leave only 6x. Option C gives the wrong dimensions and does not represent the missing region. Option D removes the main square region, which is essential. Common confusion: a middle term in a square model may be made from two equal rectangles, so do not count only one strip.