Visual models of identities

The correct answer is B. Start with the large square of area a². Removing one strip of dimensions a by b removes ab; removing the second strip removes another ab. However, the two strips overlap at the corner in a b by b square, whose area is b². That common corner was included in both removed areas, so it was subtracted twice. To correct the double subtraction, add b² back once: a² − ab − ab + b² = a² − 2ab + b² = (a − b)². A is false because the strips do overlap. C is false because a complete strip has area ab, not b². D is not generally true and does not describe the correction. Memory cue: when two removed regions overlap, add their common part back once.

The correct answer is A: (a − b)². Begin with a², the area of the large square. Two strips, each of area ab, are removed, so the temporary area is a² − ab − ab = a² − 2ab. The corner region of area b² belongs to both strips and was therefore removed twice; adding it back once gives a² − 2ab + b². By the identity (a − b)² = a² − 2ab + b², the remaining area is (a − b)². B and D both contain a positive 2ab term, which would describe adding the mixed rectangles rather than removing them. C, a² − b², is the difference-of-squares identity and has no −2ab term. The visual operation is subtraction with overlap correction, so A is unambiguous.

The correct answer is A: x² + 8x + 16. The total area of a divided figure is the sum of the areas of all its non-overlapping parts. Add the four regions: x² + 4x + 4x + 16. The two equal rectangular strips combine to give 8x, so the result is x² + 8x + 16. This also agrees with the square identity (x + 4)² = x² + 2(x)(4) + 4² = x² + 8x + 16. B includes only one 4x strip and therefore misses part of the model. C incorrectly changes the two 4x strips into 16x. D has a negative middle term, which would belong to (x − 4)², not a square of side x + 4. Memory cue: in a square-of-a-sum model, add both equal mixed rectangles.

The correct answer is C: 36. In the visual model for (p + 6)², the large square is divided into a p by p square, two rectangles with dimensions p by 6, and a small corner square with dimensions 6 by 6. The corner square therefore has area 6 × 6 = 36. A is p², the area of the variable square, not the constant corner. B is the area of one mixed rectangle, p × 6 = 6p. D is the combined area of the two mixed rectangles, 6p + 6p = 12p. The full expansion is (p + 6)² = p² + 12p + 36, so the constant term identifies the corner square. Memory cue: in (variable + constant)², the small corner square is the constant squared.

The correct answer is A: (y + 5)². Begin with the pieces of the area model. The large square has area y², so its side is y. Each strip has area 5y, which comes from side lengths y and 5. Since there are two strips, their combined area is 5y + 5y = 10y. The corner is a 5 by 5 square, so its area is 25. Therefore the complete area is y² + 10y + 25. Using the identity (a + b)² = a² + 2ab + b² with a = y and b = 5 gives (y + 5)² = y² + 10y + 25. Option A matches every part of the model. Option B would give y² − 10y + 25 and would require subtracting strips, not adding them. Option C is the square of 5y, which is 25y², and does not describe this four-part model. Option D would produce y² + 50y + 625, so both its middle and constant terms are wrong. Memory cue: two positive equal strips mean the middle term is +2ab.

The correct answer is B: −6m. Treat the square as (m − 3)(m − 3). The large m by m square contributes m². One middle rectangle has dimensions m and 3, so its magnitude is 3m; because the side is reduced by 3, this part is subtracted and contributes −3m. The second middle rectangle is identical and contributes another −3m. Together, the two middle parts contribute −3m − 3m = −6m. The corner is a 3 by 3 square with area 9, giving the complete identity (m − 3)² = m² − 6m + 9. Option A has the correct magnitude but the wrong sign: it would belong to (m + 3)². Option C counts only one middle rectangle and also has the wrong sign. Option D has the correct sign but counts only one of the two strips. The corner 9 is a constant term, not part of the middle term. Memory cue: (a − b)² has middle term −2ab.

The correct answer is B: x² + 14x + 49. Start with the original x by x square, whose area is x². The first added strip has dimensions x and 7, so its area is 7x. The second strip has the same dimensions and also has area 7x. Thus the two strips together contribute 14x. The corner is a 7 by 7 square, so its area is 49. Adding all four pieces gives x² + 7x + 7x + 49 = x² + 14x + 49. This is exactly (x + 7)². Option A includes only one strip and therefore misses 7x. Option C assigns 49 to the coefficient of x and 14 to the constant term, confusing the pieces. Option D uses negative middle terms, which would describe removing strips rather than adding them. A reliable check is to count the two identical x by 7 rectangles separately before combining them. The number of strips determines the coefficient 2 × 7 = 14.

The correct answer is A: r², 2r, 2r, and 4. A square with side r + 2 can be partitioned into a large r by r square, two rectangles each measuring r by 2, and a small 2 by 2 corner square. Their areas are r², 2r, 2r, and 4. Adding them gives r² + 2r + 2r + 4 = r² + 4r + 4, which is the expansion of (r + 2)². Option B leaves out one of the two equal strips, so it gives too little area. Option C doubles each strip incorrectly: an r by 2 strip has area 2r, not 4r. Option D uses negative strip areas, but the 2 units are added to the side, so the strips must have positive areas. The corner is 2² = 4. Memory cue: for (a + b)², the visual pieces are a², ab, ab, and b²; there are always two middle rectangles.

The correct answer is A: x + 3 and x + 5. Add the four displayed areas: x² + 3x + 5x + 15 = x² + 8x + 15. To find side lengths, look for two binomials whose product has middle coefficient 8 and constant product 15. Since 3 + 5 = 8 and 3 × 5 = 15, factorisation gives x² + 8x + 15 = (x + 3)(x + 5). Therefore the rectangle can have sides x + 3 and x + 5. Option B is based on the numbers 8 and 15 themselves, but (x + 8)(x + 15) has middle term 23x and constant 120, so it does not match. Option C gives (x − 3)(x − 5) = x² − 8x + 15, with the wrong sign on the middle term. Option D gives x² + 8x + 12, so its constant product is wrong. Memory cue: in (x+p)(x+q), p+q is the coefficient of x and pq is the constant term.

The correct answer is C: a rectangle with sides a + 4 and a − 4. Use the difference-of-squares identity (p + q)(p − q) = p² − q². Here p = a and q = 4, so (a + 4)(a − 4) = a² − 4² = a² − 16. The area model therefore has two side lengths, one increased by 4 and the other decreased by 4. Option A represents (a + 4)² = a² + 8a + 16, which contains positive middle terms and is not the given expression. Option B represents (a − 4)² = a² − 8a + 16, which also has a middle term and a positive constant. Option D has area 16a, not a² − 16. Notice that the difference-of-squares expression has no middle term because the +4a and −4a contributions cancel. Memory cue: a² − b² always factors as (a+b)(a−b), so think of a rectangle with sum and difference as its sides.

The correct answer is A. Since 8 = 10 − 2, substitute a = 10 and b = 2 into (a − b)² = a² − 2ab + b². This gives 8² = 10² − 2(10)(2) + 2². In the area picture, the original 10 by 10 square has area 10². Removing one strip of area 10·2 and another strip of area 10·2 subtracts 40 in total. Their corner overlap, a 2 by 2 square, has been subtracted twice, so 2² = 4 must be added back. The result is 100 − 40 + 4 = 64 = 8². Option B uses plus signs and represents (10 + 2)², not a reduced square. Option C forgets the two rectangular strips and the overlap correction. Option D adds a corner square without representing the removed strips. Common confusion: when two strips are removed, the overlap is counted twice and must be restored once.

The correct answer is A: use 99² = (100 − 1)². Choose the nearby round number 100 and write 99 as 100 − 1. Applying (a − b)² = a² − 2ab + b² gives 99² = 100² − 2(100)(1) + 1² = 10,000 − 200 + 1 = 9,801. This model is especially convenient because 100² and 2·100 are easy to calculate. Option B is not the useful representation asked for: although 90 + 9 equals 99, the resulting calculation is less convenient and the option does not use the nearest round-number idea. Option C gives (99 + 1)(99 − 1) = 100 × 98 = 9,800, which is 99² − 1, so it is not equal to 99². Option D gives 10,001, not 9,801, because it ignores the cross term. Memory cue: for a number just below a round number, use (N − 1)² = N² − 2N + 1.

The correct answer is B: 12x. Write the side as a + b with a = 2x and b = 3. In the square model for (a + b)², the two middle rectangles have areas ab and ba. The first is (2x)(3) = 6x, and the second is (3)(2x) = 6x. Their total is 6x + 6x = 12x, which is the middle term 2ab in the expansion (2x + 3)² = 4x² + 12x + 9. Option A gives only one of the two middle rectangles, not their total. Option C incorrectly multiplies 3 by x and misses the factor 2 in the side 2x. Option D is the area of the large 2x by 2x square, so it is the leading term, not the middle area. The 3 by 3 corner contributes 9. Memory cue: always calculate both cross-rectangles; the middle contribution is 2ab, not ab.

The direct answer is A: 9y^2. To build the area model, split the side 3y+2 into two parts, 3y and 2. The part whose length and width are both 3y is a square, so its area is (3y)(3y)=9y^2. This is the largest square-type region because 3y is the larger part of the side in the algebraic model. Option A is therefore correct. Option B, 6y, comes from the two rectangular strips: (3y)(2)+(2)(3y)=6y+6y=12y if both strips are counted; in any case it is not one square region. Thus it cannot be the requested square part. Option C is the small 2×2 corner square, whose area is 4, so it is smaller. Option D is a length, not an area, and therefore cannot represent a square-type area. Memory cue: in a square-of-a-binomial model, square each side piece first; the two variable pieces give (3y)^2=9y^2.

The direct answer is A: 2x+3. A square with side 2x+3 has area (2x+3)^2. Apply the identity (p+q)^2=p^2+2pq+q^2 with p=2x and q=3. Then p^2=(2x)^2=4x^2, 2pq=2(2x)(3)=12x, and q^2=3^2=9. Adding these parts gives 4x^2+12x+9, exactly the model in the question. Option A is correct. Option B, 4x+9, adds area values as if they were lengths; its square would not produce the given three parts. Option C combines 2x with 9, but 9 is an area, not the small side; the corresponding side is its square root, 3. Option D uses 4x and 3, although the square root of 4x^2 is 2x, not 4x. Memory cue: for a perfect-square area, take the square roots of the first and last terms, then check that the middle term is twice their product.

The direct answer is A. Use (p-q)^2=p^2-2pq+q^2 with p=2a and q=5. The large square contributes p^2=(2a)^2=4a^2. Each of the two mixed rectangular tiles has signed area -pq=-(2a)(5)=-10a, so together they contribute -20a. The small corner square contributes q^2=25. Therefore the tile group is 4a^2, -10a, -10a, and 25, and option A is correct. Option B has positive middle tiles, which would represent (2a+5)^2 and produce +20a. Option C incorrectly squares 2a as 2a^2 instead of 4a^2 and also uses the wrong mixed-term size. Option D combines the two -10a rectangles into one -20a entry; although its total expression is numerically equivalent, it is not the requested group of separate tiles. Memory cue: for (p-q)^2, the two mixed tiles are negative, while the two square tiles remain positive.

The direct answer is B: 9x. Multiply the two side expressions using the distributive property: (x+2)(x+7)=x·x+x·7+2·x+2·7=x^2+7x+2x+14=x^2+9x+14. In the visual model, x^2 is the large x-by-x square, 7x and 2x are the two rectangular strips, and 14 is the small corner rectangle. The middle term is the total of the strips, 7x+2x=9x. Option A, 14x, wrongly multiplies 2 and 7 and then attaches x; 2×7 actually gives the constant corner 14. Option C is the main square term, not the middle term. Option D is the product of the constant parts and is the constant term. Memory cue: in (x+a)(x+b), the middle term is (a+b)x, while the corner is ab.

The direct answer is A: the sides are x+6 and x+4. In an area model, the x^2 tile comes from x·x. A strip of area 6x is formed by x·6, so one side contributes the constant part 6. Similarly, the 4x strip comes from x·4, giving the other side constant part 4. Thus the dimensions are (x+6) and (x+4). Their product verifies the model: (x+6)(x+4)=x^2+4x+6x+24=x^2+10x+24. Option B incorrectly uses the sum 6+4=10 as one side and 24 as the other constant, confusing the middle and final terms. Option C has constants whose product is 24, but its strips would be 2x and 12x, not 6x and 4x. Option D gives negative strips and does not match the positive model. Memory cue: each x-strip directly reveals one added constant in a factor.

The direct answer is A: the strips are 5x and 6x. For a rectangle model of x^2+11x+30, the two constant numbers must satisfy two conditions: their product must be 30, because they form the corner, and their sum must be 11, because their x-rectangles combine to make the middle term. For 5 and 6, 5×6=30 and 5+6=11. Therefore the strips are 5x and 6x, and (x+5)(x+6)=x^2+11x+30. Option B has product 3×10=30, but its sum is 13, so it would produce 13x. Option C has the correct product 30, but its sum is 31, not 11. Option D has product 2×15=30, but its sum is 17. Memory cue: for x^2+bx+c, search for two numbers whose product is c and sum is b; both tests are necessary.

The direct answer is A: remove the smaller b-by-b square from the larger a-by-a square. A square with side a has area a^2, and a square with side b has area b^2. If the smaller square is cut out of the larger one, the remaining region has area a^2-b^2. This is the most direct visual meaning of subtraction of squares. Option B adds the two areas and therefore represents a^2+b^2, not their difference. Option C describes strips but does not by itself show either square area or their subtraction. Option D shows only an ab rectangle, whose area is ab; it does not represent a^2-b^2. The remaining shape can also be rearranged to demonstrate (a-b)(a+b), but the first visual step is the cut-out. Memory cue: a minus sign in an area model means a region is removed; for a^2-b^2, think of a smaller square cut from a larger square.

The direct answer is A: the side is x+9. Compare the expression with (x+r)^2=x^2+2rx+r^2. The constant term is 81, so r^2=81 and the positive length is r=9. Check the middle term: 2·x·9=18x, exactly as required. Therefore x^2+18x+81=(x+9)^2, so a square with side x+9 produces the given area. Option B is wrong because (x+18)^2=x^2+36x+324, giving different middle and constant terms. Option C treats 81 as a side addition, but 81 is the square of the added part, not the part itself. Option D gives (x-9)^2=x^2-18x+81, whose middle term is negative. A useful method is to take the square root of the constant term and then verify the middle term. Memory cue: a positive middle term in a perfect square usually indicates the plus form, while a negative middle term indicates the minus form.

The direct answer is A. Add the four regions in the complete square: the first small square has area a^2, the second has area b^2, and the two rectangles contribute ab+ab=2ab. Thus the total area is a^2+2ab+b^2. The outer side is made by joining lengths a and b, so it is a+b; its area is (a+b)^2. Therefore (a+b)^2=a^2+2ab+b^2. Option B has -2ab and would require the mixed rectangles to be negative or arranged for subtraction, which is not the model described. Options C and D are equivalent forms of the difference-of-squares identity and involve subtraction, not two positive ab rectangles filling one complete square. Memory cue: two positive ab rectangles mean +2ab; two negative mixed parts mean -2ab.

The direct answer is B: the area is a^2-9. Rearranging pieces does not change the total area, so calculate the original rectangle area by multiplying its side lengths: (a+3)(a-3). Use the difference-of-squares identity (p+q)(p-q)=p^2-q^2, with p=a and q=3. This gives a^2-3^2=a^2-9. Option A has the wrong plus sign; the opposite signs in the two factors require subtraction. Option C is (a-3)^2=a^2-6a+9, which would be the area of a square with side a-3, not the given rectangle. Option D is (a+3)^2 and likewise describes a different square. Memory cue: matching brackets with opposite signs, such as (a+3)(a-3), cancel the middle terms and leave a^2-3^2.

The direct answer is A. A square with side 5x has area (5x)^2=25x^2, and a square with side 6 has area 6^2=36. Therefore 25x^2-36 means subtracting the area of the smaller square from the area of the larger square. This is the visual meaning of a difference of two squares: a^2-b^2=(a+b)(a-b). Option A matches both areas and sides. B is wrong because a square of side 25x has area 625x^2, not 25x^2; it confuses a side with an area. C represents (5x+6)^2, which includes positive cross terms. D represents (5x-6)^2, which also includes a middle term and is not a difference of two square areas. Memory cue: first take square roots of the two terms, then draw one square being removed from the other.

The direct answer is B, 4x. The area of a square is side multiplied by itself, so the two areas are (x+1)^2 and (x-1)^2. Subtract them: (x+1)^2-(x-1)^2=(x^2+2x+1)-(x^2-2x+1)=4x. The same result comes quickly from a^2-b^2=(a-b)(a+b): here a-b=2 and a+b=2x, so the product is 4x. A is only half the result and misses one of the two cross contributions. C is (x+1)(x-1), not the difference of the two areas. D would be the answer only if x=1, not generally. A useful warning is that side differences and area differences are not the same; square each side before subtracting.