Visual models of identities

Correct answer: D. 6 and 7

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.

Correct answer: B. 4x² − 9

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.

Correct answer: A. One p² square, one q² square, and two rectangles of area pq

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.

Correct answer: D. A square with side (7a−5b)

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.

Correct answer: D. 7x and 8x, the two strips

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.

Correct answer: C. 4, four 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.

Correct answer: B. 16x, the complete border

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.

Correct answer: A. 3x²+x−10, signed sum

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.

Correct answer: A. (x^2+xy+y^2) / one rectangle partial

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.

Correct answer: A. (a+b)^2=a^2+2ab+b^2 / two strips and corner

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.

Correct answer: A. (5x-1) and (5x+1) / sides

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.

Correct answer: A. 7xy / each middle rectangle

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.

Correct answer: B. 8ab

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.

Correct answer: A. 8x / remaining area

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.

Correct answer: A. (3x + 2)(5x − 4), with middle areas 10x and −12x

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.

Correct answer: A. 4c(a + b), the strip area

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.

Correct answer: A. (a+b)² = a² + 2ab + b²

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.

Correct answer: D. m + 9

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.

Correct answer: B. 11r and −11r

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.

Correct answer: C. 9x² − 16

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.

Correct answer: D. 6 and 9

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

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

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)

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

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.