Visual models of identities

The direct answer is A. Split each side of the square into lengths x and 10. The resulting four rectangular regions have areas x times x=x^2, x times 10=10x, 10 times x=10x, and 10 times 10=100. Their sum is x^2+10x+10x+100=x^2+20x+100, which is (x+10)^2. B combines the two equal strips into one expression, so it does not list four separate regions as asked. C incorrectly gives one strip as 100x, even though one dimension of that strip is 10. D makes both strips too large and gives the corner the wrong value. Memory cue: in a square split into x and 10 on both sides, expect one x^2 region, two identical 10x regions, and one 10^2 corner.

The direct answer is A. Add the vertical parts to get the total height x+8, and add the horizontal parts to get the total width x+2. The rectangle’s area is therefore (x+8)(x+2). Multiply each part: x times x=x^2, x times 2=2x, 8 times x=8x, and 8 times 2=16. Adding gives x^2+2x+8x+16=x^2+10x+16. B incorrectly swaps the coefficient of x and the constant term. C uses 6x, although the two cross-products add to 10x. D lists only the original boundary parts and omits the complete multiplication. A reliable check is to add the four small rectangle areas rather than trying to guess the expanded expression.

The direct answer is C: 4 and 9. In a rectangle or tile factor model, the two constant side lengths must multiply to the corner area and add to the coefficient of x. Here 4 times 9=36 and 4+9=13, so x^2+13x+36 factors as (x+4)(x+9). A has the correct product but the wrong sum, 37. B also has product 36, but its sum is 20, not 13. D has sum 12 and product 36, so it fails the sum condition as well. The corner alone is not enough to decide; both the product and the middle-term sum must match. Memory cue: corner gives multiplication, while the two side strips together give the x-coefficient.

The direct answer is A. Since 25=5^2, the expression is x^2-5^2, a difference of squares. The identity a^2-b^2=(a+b)(a-b) gives x^2-25=(x+5)(x-5). In the visual model, remove a 5-by-5 square from an x-by-x square. The remaining pieces can be cut and rearranged into a rectangle whose sides are x+5 and x-5. B is wrong because a sum or perfect-square model describes expressions such as (x+5)^2, which has a positive middle term. C shows addition rather than removing a smaller square, and it does not produce the required unequal factors. D leaves only x^2 and does not represent the subtraction of 25. Common warning: difference of squares is not the same as (x-5)^2.

The direct answer is A, 3a+4. Recognise the first term as (3a)^2 and the last term as 4^2. For a perfect-square expression, the middle term must be twice the product of these parts: 2(3a)(4)=24a. Therefore 9a^2+24a+16=(3a+4)^2, so the visual square has side 3a+4. B and D use 9a as though it were the side, but 9a^2 has square root 3a, not 9a. C gives a side whose cross-term would be 2(3a)(8)=48a, so it cannot match 24a. The side is obtained by taking the square roots of the two corner square areas and combining them with the sign of the middle term. Memory cue: check first square, last square, then twice the product.

The direct answer is B, -8x. Split the side into 4x and -1. When the square is expanded, the two cross-rectangles have areas (4x)(-1)=-4x each. Their combined contribution is -4x-4x=-8x. Equivalently, use (a-b)^2=a^2-2ab+b^2 with a=4x and b=1: (4x-1)^2=16x^2-8x+1. A has the correct magnitude but the wrong sign; the negative part of the side makes both cross-products negative. C counts only one cross-rectangle and misses the second. D is the first square term, not the middle term. Memory cue: in (a-b)^2, the middle term is always -2ab, while in (a+b)^2 it is +2ab.

The direct answer is B. The x^2 tile shows that both factors begin with x. The two side tiles are -3x and -7x, so the constant parts of the factors are -3 and -7. Their product is (-3)(-7)=21, matching the corner tile, and their sum gives -3x-7x=-10x. Thus (x-3)(x-7)=x^2-10x+21, exactly the collection shown. A would produce positive side terms, +3x and +7x, so its strips have the wrong signs. C has factors that do not produce the listed two strips or the correct corner. D gives one positive and one negative strip and a negative corner, not +21. Sign cue: two negative side lengths produce a positive corner because negative times negative is positive.

The direct answer is A. The x^2 square supplies the main x-by-x region. Removing a 4x strip reduces one side by 4, and removing a 9x strip reduces the other side by 9. The missing corner has area 4 times 9=36, so adding that corner completes the signed rectangle. Its side lengths are therefore x-4 and x-9, giving (x-4)(x-9)=x^2-13x+36. B would correspond to adding both strips, producing positive middle terms rather than removing them. C does not use the two separate strip widths as the factors and has an unrelated constant side. D incorrectly assumes the two removed widths are equal; 4 and 9 are different. Common-confusion warning: the corner 36 confirms 4*9, but it does not make the factors x-6 and x-6.

Direct answer: A, because x² − 2x − 15 = (x + 3)(x − 5). An area model represents the two factors as the complete side lengths of a rectangle. To find the numbers, look for two integers whose product is −15 and whose sum is −2. The numbers 3 and −5 satisfy both conditions: 3 × (−5) = −15 and 3 + (−5) = −2. Therefore the four model areas are x², 3x, −5x, and −15; their sum is x² + 3x − 5x − 15 = x² − 2x − 15. Option A is correct because its expansion has exactly the required middle term and constant term. Option B gives (x − 3)(x + 5) = x² + 2x − 15, so the middle sign is wrong. Option C gives x² − 16x + 15, so both the middle and constant terms are wrong. Option D gives x² − 13x − 30, so its product is not −15. Memory cue: for x² + bx + c, check the pair by sum b and product c, including signs.

Direct answer: A. The required difference is (x + 6)² − (x − 6)². Let a = x + 6 and b = x − 6. The difference-of-squares identity gives a² − b² = (a − b)(a + b). Here, a − b = 12 and a + b = 2x, so the total difference is 12 × 2x = 24x. In a rearranged visual model, this difference can be represented by four congruent rectangular strips, each measuring 6 by x and therefore having area 6x. Their total area is 4 × 6x = 24x. Option A matches both the picture and the calculation. Option B also totals 24x, but it describes eight strips of 3x, which is not the stated model. Option C totals 144, not 24x. Option D totals 24x numerically but describes only two strips of 12x, not the intended four-strip arrangement. Memory cue: when two square areas are subtracted, first try a² − b² = (a − b)(a + b).

Direct answer: A. A square with side a + b can be divided into four parts: one square of area a², one square of area b², and two rectangular regions. Each rectangular region has dimensions a and b, so each has area ab. Thus (a + b)² = a² + ab + ab + b² = a² + 2ab + b². The two middle rectangles are equal because the large figure is a square and the same two lengths occur in both directions. Option A guarantees the pair of equal ab strips. In option B, the two cross-rectangles have areas ab and ac, which are not generally equal. In option C, the corresponding areas are −ab and −ac in a signed model; they are not necessarily equal either. Option D represents a difference of squares, usually by cutting and rearranging pieces, and it does not necessarily contain two equal rectangular strips. Memory cue: the middle term 2ab in (a + b)² comes from two equal ab rectangles.

Direct answer: B. A square with side x + 5 contains an x by x square, two rectangles measuring x by 5, and a 5 by 5 square. Their areas are x², 5x, 5x, and 25. Therefore (x + 5)² = x² + 5x + 5x + 25 = x² + 10x + 25. The student included x² and 25 but omitted both middle rectangles. Option A gives only one of the two strips, so it is incomplete. Option B correctly gives their combined area, 5x + 5x = 10x. Option C is not produced by multiplying the relevant side lengths; the small square has area 25, not 25x. Option D incorrectly counts the x by x square twice and does not describe the missing strips. The common error is to think that (x + 5)² equals x² + 25. Memory cue: a square of a sum always includes two cross-products, giving 2ab.

Direct answer: A. Begin with a large square of area x². To represent side x − 8, remove a vertical strip of width 8 and a horizontal strip of width 8. The vertical strip has area 8x and the horizontal strip also has area 8x. However, the 8 by 8 corner belongs to both strips, so it has been removed twice. To correct this double removal, add that corner back once. Its area is 8 × 8 = 64. The identity is therefore (x − 8)² = x² − 8x − 8x + 64 = x² − 16x + 64. Option A explains the purpose of the added square. Option B is wrong because the 64 does not enlarge the original x²; it repairs the counting. Option C is wrong because the middle term comes from the two strips, giving −16x. Option D confuses area with boundary length; the small square does not represent perimeter. Memory cue: in (a − b)², subtract two ab regions but add b² back once.

Direct answer: A. Use the identity (a + b)² = a² + 2ab + b² with a = 50 and b = 2. A visual square of side 52 is divided into a 50 by 50 square, two rectangles each measuring 50 by 2, and a 2 by 2 square. Their areas are 2500, 100, 100, and 4. Adding them gives 2500 + 200 + 4 = 2704, which is 52². Option A includes all four regions and therefore gives the correct value without ordinary long multiplication. Option B leaves out the two rectangles and gives only 2504. Option C uses the subtraction identity and gives the wrong sign for the cross-term. Option D calculates a difference of squares, not the square of a sum, so it is a different expression entirely. Memory cue: for a sum, the two middle rectangles are positive and must not be omitted.

Direct answer: A. The complete horizontal side is x + 4 and the complete vertical side is x − 6, so the rectangle’s area is (x + 4)(x − 6). In an area model, multiply every top label by every left label: x·x = x², x·(−6) = −6x, 4·x = 4x, and 4·(−6) = −24. Add the four signed areas: x² − 6x + 4x − 24 = x² − 2x − 24. Option A is therefore correct. Option B would incorrectly add 6x instead of using −6x. Option C has the wrong signs for the middle and constant terms. Option D has the correct constant term but the wrong middle coefficient. A negative label means a negative signed area in the algebraic model; it is not permission to ignore the sign. Memory cue: multiply each row and column pair separately, then combine like terms.

Direct answer: A. A large square with side a + b has area (a + b)². Divide each side into parts a and b. The figure then contains an a by a square of area a², a b by a rectangle of area ab, another equal rectangle of area ab, and a b by b square of area b². Adding the four parts gives a² + ab + ab + b² = a² + 2ab + b². Thus the entire collection is one large square whose side is a + b. Option A matches the outer shape and the identity. Option B gives (a − b)² = a² − 2ab + b², so its middle term is negative. Option C gives (a + b)(a − b) = a² − b², which is a difference of squares. Option D has area ab only, not the full expression. Memory cue: the plus-plus square identity corresponds to the outside side a + b.

Direct answer: B. Keep the two expressions separate. The difference of squares is a² − b², while the square of a difference is (a − b)² = a² − 2ab + b². To change a² − b² into a² − 2ab + b², subtract the net amount (2ab − b²): (a² − b²) − (2ab − b²) = a² − b² − 2ab + b² = a² − 2ab. This wording concerns the signed region that must be removed from the original difference-of-squares arrangement; equivalently, the target expression contains the additional signed contribution −2ab + b² relative to a². Option B gives the positive amount removed. Option A states the signed contribution added in the target expression, not the positive area removed. Option C removes both original squares and destroys the expression. Option D removes only one ab strip and is incomplete. Memory cue: a² − b² and (a − b)² are different identities; the latter contains −2ab + b².

Direct answer: A. Compare the expression with the perfect-square identity (x + k)² = x² + 2kx + k². Since the middle term is 6x, we have 2k = 6 and therefore k = 3. The visual model has one x by x square, two equal rectangles measuring x by 3, and one 3 by 3 square. Each rectangular strip has area x × 3 = 3x. Together, the two strips contribute 3x + 3x = 6x, which is the middle term. Option A asks for one strip and is correct. Option B is the combined area of both strips, not each one. Option C multiplies unlike dimensions and does not represent a model part. Option D is a length, x + 3, rather than an area. Memory cue: in a perfect square, split the middle coefficient equally; 6x becomes 3x + 3x.

The correct answer is A. Split one side into 2x and 1, and split the other side into x and 3. Each small rectangle is found by multiplying one horizontal part by one vertical part: (2x)(x) = 2x², (2x)(3) = 6x, (1)(x) = x, and (1)(3) = 3. Adding them gives 2x² + 6x + x + 3 = 2x² + 7x + 3, exactly the stated area. B incorrectly uses 3x instead of (2x)(3) = 6x. C incorrectly uses 2x instead of (1)(x) = x. D incorrectly changes (2x)(x) into x². Thus A is confirmed both by multiplication and by the sum of the cells. Memory cue: label the row and column parts, then multiply across each grid cell.

The correct answer is A. The entire figure is a square whose side is a + b, so its total area is (a + b)². The four non-overlapping regions have areas a², ab, ab, and b². Adding them gives a² + ab + ab + b² = a² + 2ab + b². Therefore (a + b)² = a² + 2ab + b². B has negative mixed terms and represents the square of a difference. C is the difference-of-squares identity; its factors form a rectangle and its mixed terms cancel. D omits the two ab regions, so it is not generally true. For positive a and b, the omitted 2ab is a real part of the total area. Memory cue: the square of a sum has two equal mixed rectangles, so its middle term is +2ab.

The direct answer is A. The larger square has area (x+4)² and the smaller has area (x+2)². Subtract: (x+4)²−(x+2)²=(x²+8x+16)−(x²+4x+4)=4x+12. The x² terms cancel because both squares contain the same central x-by-x part. B is only half of the correct difference and does not represent the full area change. C is the product (x+2)(x+4), not a subtraction of square areas. D keeps only the constant change and ignores the strips involving x. A useful visual cue is that enlarging a square from side x+2 to x+4 adds strips whose total area depends on x as well as the corner area.

The direct answer is A: a square with side x+y. Start with the identity (x+y)²=x²+2xy+y². In an area model, x² is an x-by-x square, y² is a y-by-y square, and 2xy is made from two rectangles, each having area xy. These four positive regions fit together along two lengths x and y, so the outside length in each direction is x+y. Therefore the complete figure is a square of side x+y. B represents (x−y)²=x²−2xy+y², where the mixed part has a negative sign. C represents (x+y)(x−y)=x²−y², so the mixed terms cancel. D has area only xy and cannot contain all four regions. Memory cue: a positive 2xy means two mixed rectangles are added.

The direct answer is A. The main square has area 16p²=(4p)², so its side is 4p. The corner square has area 25=5², so its side is 5. In the model, the outer side is the sum of these adjacent lengths: 4p+5. This also checks by the identity (4p+5)²=16p²+40p+25, where the two strips contribute 20p+20p=40p. B adds areas as if they were lengths. C doubles the strip width and therefore gives the wrong outer side. D incorrectly uses 8p instead of the square root 4p. Memory cue: when a square area is given, take its square root to recover the side before adding lengths.

The direct answer is A. Start with an a-by-a square of area a² and remove a b-by-b square of area b². The remaining L-shaped pieces together have area a²−b². After cutting and rearranging those pieces, one side becomes a+b, while the other becomes a−b. Its area is therefore (a+b)(a−b), and expanding gives a²−b² because the +ab and −ab contributions cancel. B would have area (a−b)², not the required difference. C would have area (a+b)². D has area ab and does not match. The condition a>b ensures a−b is a positive length. Memory cue: difference of squares always factors into conjugate factors.

The direct answer is A. Divide each side of the large square into lengths a and b. The resulting four regions are an a-by-a square, an a-by-b rectangle, a b-by-a rectangle, and a b-by-b square. The two mixed rectangles have areas a×b=ab and b×a=ab. They are different positions in the model, although their numerical areas are equal, so together they contribute 2ab. B is wrong because the a² square is not what creates the mixed rectangles. C reverses the containment relationship; the b² square is a separate corner region. D omits both mixed regions and gives an incomplete total. Memory cue: the two cross-products in (a+b)(a+b) are ab+ab.