Visual models of identities

The direct answer is C: (x+9)²=x²+18x+81. Start with the square identity (x+a)²=x²+2ax+a². The small square has area 81, so its side is √81=9. The two equal rectangular strips each have area 9x, and together they contribute 9x+9x=18x. Therefore the complete area is x²+18x+81, which is (x+9)². Option A uses 18 as the small side, producing a small square of area 324 and a middle term 36x. Option B has the correct small-square area and magnitude of the middle term, but its negative sign represents (x-9)², not the positive 18x shown. Option D is the difference-of-squares identity and has no 18x middle term. Memory cue: half of the middle-term coefficient gives the side of each equal strip; 18÷2=9.

The direct answer is D: (a-b)(a+b). The original large square has area a², and the removed small square has area b². Therefore the remaining area is a²-b². The difference-of-squares identity says a²-b²=(a-b)(a+b), so when the remaining pieces are rearranged into a rectangle, its side lengths are a-b and a+b. Option A, (a-b)², would describe a square whose two sides are both a-b; it is not the difference between two square areas. Option B adds the two areas even though one square was removed. Option C, 2ab, is the total of the two cross-rectangle areas in (a+b)², not the remaining area here. Memory cue: removing one square from another means subtract first; a difference of squares factors into the sum and difference of the sides.

The direct answer is A, 891. The factors 27 and 33 are equally distant from 30: one is \(30-3\) and the other is \(30+3\). Use the difference-of-squares identity \((u-v)(u+v)=u^2-v^2\). Thus \((30-3)(30+3)=30^2-3^2=900-9=891\). A direct check also gives \(27\times33=27(30+3)=810+81=891\). Option B, 900, is only \(30^2\); it forgets to subtract the small square \(3^2\). Option C, 909, incorrectly adds 9 instead of subtracting it. Option D, 873, uses an incorrect subtraction and does not follow the identity. The visual model shows a large 30-by-30 square with a 3-by-3 correction removed. Memory cue: numbers equally spaced around a middle number give middle-square minus distance-square.

The direct answer is A, \(4x^2-9\). A rectangle’s area is length multiplied by breadth, so calculate \((2x+3)(2x-3)\). These factors have the form \((u+v)(u-v)=u^2-v^2\), with \(u=2x\) and \(v=3\). Therefore the area is \((2x)^2-3^2=4x^2-9\). Expanding confirms it: \(2x(2x-3)+3(2x-3)=4x^2-6x+6x-9=4x^2-9\); the middle terms cancel. B has the wrong plus sign and would represent adding the two squares. C has the wrong leading coefficient because \((2x)^2=4x^2\), not \(2x^2\). D is \((2x-3)^2\), the square of one factor, not the product of the two different factors. Memory cue: plus and minus around the same two terms produce a difference of squares.

For \((3x+2)^2=(3x+2)(3x+2)\), the \(x\)-term parts come from \(3x\times2\) and another \(3x\times2\). Each rectangle has area \(3x\cdot2=6x\). Two such rectangles give \(6x+6x=12x\).

The direct answer is C, 6. Divide each side of the square into three labelled lengths, \(a\), \(b\), and \(c\). This creates a 3-by-3 grid. The three diagonal cells are squares with areas \(a^2\), \(b^2\), and \(c^2\). Every off-diagonal cell is a rectangle: the pairs are \(ab\) and \(ba\), \(ac\) and \(ca\), and \(bc\) and \(cb\). Thus there are six rectangular cross regions in all. A counts only the three pair types and ignores their two orientations. B misses two regions, while D counts all nine cells and incorrectly includes the three diagonal squares. The area identity represented is \((a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\). Memory cue: in an n-by-n grid, the off-diagonal count is \(n(n-1)\); for n=3 it is 6.

The direct answer is B. The area of the large square is side multiplied by itself: \((a+b+c)^2\). In the visual grid there are three diagonal squares, with areas \(a^2\), \(b^2\), and \(c^2\). There are also six off-diagonal rectangles: two of area \(ab\), two of area \(bc\), and two of area \(ca\). Adding all parts gives \(a^2+b^2+c^2+2ab+2bc+2ca\). A includes only one rectangle for each pair and therefore misses half of the cross area. C uses negative cross terms, which would correspond to a different arrangement involving subtraction. D introduces \(abc\), a cubic term, but a square area from these side lengths has total degree two and contains no triple product. Memory cue: every pair among a, b, c appears twice in the grid.

The direct answer is A. The entire square has side \(p+q\), so its area is \((p+q)^2\). The partition divides that area into four parts: one \(p^2\) square, one \(q^2\) square, and two rectangles each of area \(pq\). Adding them gives \(p^2+pq+pq+q^2=p^2+2pq+q^2\). Therefore \((p+q)^2=p^2+2pq+q^2\). Option B describes a square of side \(p-q\), where the cross terms are negative, not the given all-positive partition. Option C represents a difference of two squares and would require subtraction and factors \(p+q\) and \(p-q\). Option D is false: \((p+q)(p-q)=p^2-q^2\), not \(p^2+q^2\). Memory cue: two equal \(pq\) rectangles create the middle term \(2pq\).

Direct answer: Option D, 16, is correct. In the area model for a square with side 3x+4, each side is split into a variable part 3x and a constant part 4. The small corner formed by the two constant lengths has dimensions 4 by 4. Its area is therefore 4×4=4²=16. This is called the constant square because it contains no x. Option A is only the side length, not the area. Option B is twice the side length and does not describe a square area. Option C is the product of 3 and 4, which relates to a mixed rectangle rather than the constant corner square. Option D correctly identifies the contribution of the corner region. In a model of (3x+4)², the four-part expansion is 9x²+24x+16; the final constant term comes from this 4×4 square.

Direct answer: Option C, 98, is correct. Divide the square of side x+y into one x by x square, one y by y square, and two x by y rectangles. The two square regions together have area x²+y²=50. The two rectangular regions together have area 2xy=48. The entire square is made from all four non-overlapping regions, so its area is 50+48=98. Equivalently, (x+y)²=x²+2xy+y². Option A, 2, is just the number of rectangular pieces and is not an area. Option B counts only the rectangles and omits the square pieces. Option D multiplies 50 and 48, even though the regions must be added to obtain the total area. The important idea is that a partitioned figure’s total area equals the sum of the areas of all its parts.

Answer: A. Imagine a large square whose side is split into lengths \(a\) and \(b\) in both directions. The \(a\)-by-\(a\) corner is a square of area \(a^2\), and the \(b\)-by-\(b\) corner is a square of area \(b^2\). The remaining two corners each have side lengths \(a\) and \(b\), so each is an \(ab\) rectangle. Their combined area is \(ab+ab=2ab\). Adding every region gives \(a^2+2ab+b^2=(a+b)^2\). B misses one cross rectangle and therefore gives only \(a^2+ab+b^2\). C has the wrong number of \(a^2\) squares. D uses incorrect shapes and areas. Memory cue: a binomial square has two end squares and two identical cross rectangles.

Answer: A. The entire figure is a square with side \(a+b\), so its total area is \((a+b)^2\). The four stated regions have areas \(a^2\), \(ab\), \(ab\), and \(b^2\). Adding them gives \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Equating the total area with the sum of the parts produces \((a+b)^2=a^2+2ab+b^2\). B describes a square with subtraction and therefore has a negative middle term. C and D are equivalent forms of the difference-of-squares identity; they involve factors \(a+b\) and \(a-b\), not four regions with two positive \(ab\) parts. Memory cue: two positive \(ab\) regions mean the middle term is \(+2ab\).

Answer: A. Split each side of the large square of side \(a+b\) into lengths \(a\) and \(b\). The four resulting regions are an \(a\)-by-\(a\) square with area \(a^2\), a \(b\)-by-\(b\) square with area \(b^2\), and two rectangles measuring \(a\)-by-\(b\), each with area \(ab\). Thus the total is \(a^2+ab+ab+b^2=(a+b)^2\). B leaves out one rectangle, so its total is too small. C describes a different rectangle and is related to the difference of squares. D counts two \(a^2\) squares that do not arise from this partition. Memory cue: splitting both directions creates four pieces: square, rectangle, rectangle, square.

Answer: B, 12. A perfect-square trinomial has the form \((x+b)^2=x^2+2bx+b^2\). The constant term is 36, so \(b^2=36\), giving \(b=6\) or \(-6\). Because the coefficient \(a\) is positive, choose \(b=6\), so the expression is \((x+6)^2\). Expanding gives \(x^2+12x+36\), hence \(a=12\). A is only the square root of the constant term, not the middle coefficient. C and D do not satisfy the perfect-square pattern: the middle coefficient must be twice 6, not three times or six times 6. Memory cue: in \(x^2+ax+b^2\), the middle coefficient of a square is twice the side constant.

Answer: A. The area of a square is the square of its side, so we calculate \((5a-2b)^2\). Use \((p-q)^2=p^2-2pq+q^2\), with \(p=5a\) and \(q=2b\). The first square is \((5a)^2=25a^2\), the cross term is \(-2(5a)(2b)=-20ab\), and the last square is \((2b)^2=4b^2\). Therefore the expansion is \(25a^2-20ab+4b^2\). B has the wrong sign because subtraction gives a negative middle term. C forgets the factor 2. D squares neither term correctly. Memory cue: square each end and use twice the product in the middle; the sign follows the sign between the terms.

Direct answer: B is correct. Start with a large square whose side is a, so its area is a^2. If a smaller square whose side is b is removed from it, the removed area is b^2. The part left over therefore has area a^2-b^2. This is different from the complete square of side a+b, whose parts are a^2, two rectangles of area ab, and b^2, giving a^2+2ab+b^2. Option A describes addition, so it produces the plus identity, not a difference. Option B correctly describes subtraction of one square from another. Option C adds two ab rectangles and therefore belongs to the middle-term construction of (a+b)^2. Option D describes the expansion of (a-b)^2, which equals a^2-2ab+b^2; it is not the direct large-square-minus-small-square model asked here. Memory cue: “large square minus small square” means a^2-b^2; “two strips” usually signals the middle term.

The governing identity is (x + a)(x + b) = x² + (a + b)x + ab. Comparing this general expansion with x² + 9x + 20 gives two conditions that must hold at the same time: a + b = 9 and ab = 20. The pair 4 and 5 satisfies both conditions because 4 + 5 = 9 and 4 × 5 = 20. Substitution verifies the result: (x + 4)(x + 5) = x² + 5x + 4x + 20 = x² + 9x + 20. Hence option A is correct. The pair 2 and 10 has the correct product but sum 12; 1 and 20 has sum 21; and 3 and 6 has sum 9 but product 18. Thus no other option matches both coefficients.

Direct answer: B, 12x. Compare the expression with the identity (x-6)^2=x^2-2(6)x+6^2=x^2-12x+36. In the area model, the two equal rectangles come from the products x×6 and 6×x. Each rectangle therefore has area 6x. Adding the two equal areas gives 6x+6x=12x, which is the negative middle term in magnitude; the minus sign indicates removal, while the question asks for total area, so we state 12x rather than -12x. Option A gives only one rectangle, not both. Option B counts both rectangles correctly. Option C is 36, the area of the small 6-by-6 square, not the removed strips. Option D doubles 12x again and has no matching pair of rectangles. A useful check is that the coefficient of x in the middle term is twice the side of the small square: 2×6=12.

Direct answer: B, 12x. A square with side 2x+3 can be split into lengths 2x and 3 along each direction. The two cross-shaped rectangular parts have dimensions 2x by 3 and 3 by 2x. Each therefore has area (2x)(3)=6x. Since there are two equal rectangles, their combined area is 6x+6x=12x. This is the middle term in (2x+3)^2: (2x)^2+2(2x)(3)+3^2=4x^2+12x+9. Option A is only the area of one rectangle. Option B correctly includes both. Option C is (3x)^2-like thinking and does not represent either rectangle. Option D is 4x, not the product of the two rectangle side lengths. The key idea is that the middle term in a binomial square comes from two equal rectangles, not from the two square corners. Memory cue: calculate one strip, then double it.

Direct answer: A, 3x+2. In a square model, the large square area is the square of one part of the outer side. Since 9x^2=(3x)^2, that part of the side is 3x. The small square has area 4=2^2, so its side is 2. The outer side is formed by placing these lengths together: 3x+2. The two rectangles confirm the result because each has area (3x)(2)=6x, exactly as stated. Option A satisfies all three pieces and expands to (3x+2)^2=9x^2+12x+4. Option B adds areas rather than taking their square roots, so it is not a length. Option C uses 4 instead of the side 2 of the small square. Option D uses 6x, which is a rectangle area coefficient, not a side length. Memory cue: in a square tile model, take square roots of the two corner-square areas, then add the resulting side lengths.

Direct answer: D, 5x. Compare the expression with (x+5)^2=x^2+2(5)x+25=x^2+10x+25. In the tile model, the x^2 tile and the 25 tile are the two square pieces. The middle term 10x is made from two equal rectangles, so each rectangle must have half of that total area: 10x÷2=5x. Equivalently, each rectangle has dimensions x and 5, giving area 5x. Option A assigns the whole middle term to each rectangle and would incorrectly produce 20x in total. Option B confuses the constant square area 25 with a rectangle area. Option C does not account for the coefficient 5 required by the side length. Option D correctly matches both the identity and the tile dimensions. A useful habit is to split the middle term into two equal parts before identifying the rectangles: 10x=(5x)+(5x).

Direct answer: A. The area of a square is side multiplied by itself, so the required expression is (3p-4)^2. Apply (u-v)^2=u^2-2uv+v^2 with u=3p and v=4. The first square is (3p)^2=9p^2. The two equal negative rectangles together have area -2(3p)(4)=-24p. The small square has area 4^2=16. Adding these signed parts gives 9p^2-24p+16. Option B misses the factor 2 and therefore gives the wrong middle coefficient. Option C treats (3p)^2 as 3p^2 instead of 9p^2. Option D has the wrong sign; subtraction in the side creates a negative middle term. The area model shows one large positive square, two negative rectangles, and one positive small square. Memory cue: square each end and use twice the product in the middle, keeping the sign from the binomial.

Direct answer: C. This is a difference-of-squares model. The large square has area 16x^2=(4x)^2, so its side is 4x. The removed square has area 25=5^2, so its side is 5. The remaining area is therefore (4x)^2-5^2. Using u^2-v^2=(u+v)(u-v), it rearranges into a rectangle with dimensions (4x+5) and (4x-5). Option A uses 25 itself as a side instead of taking its square root, so it confuses area with length. Option B uses 16x, the coefficient before squaring, rather than the side 4x. Option C correctly takes both square roots and applies the identity. Option D uses 2x, whose square is 4x^2, not 16x^2. A quick verification is to multiply C: (4x+5)(4x-5)=16x^2-25, exactly the remaining area.

The correct answer is 7. In the area model for (x+a)^2, the small corner square has side a, so its area is a^2. The given area gives a^2=49. Since the side length is normally taken as positive in this model, a=7. We can confirm this using the two rectangular strips: their combined area is ax+ax=2ax. The question gives 14x, so 2ax=14x. For a nonzero x, divide by 2x to get a=7. Thus both pieces of information agree. Option A is wrong because 14 is the coefficient of 2a, not a. Option B confuses the area 49 with the side length. Option C does not satisfy either a^2=49 or 2a=14. Option D gives the required side and coefficient. Memory cue: in (x+a)^2, the corner square is a^2 and the two strips together are 2ax.

The correct answer is A. A rectangle for (x+3)(x+8) is divided according to the two terms in each factor. Multiply each part of the first factor by each part of the second: x times x gives x^2; x times 8 gives 8x; 3 times x gives 3x; and 3 times 8 gives 24. These are the four regions in the visual model, and their sum is x^2+8x+3x+24=x^2+11x+24. Option B incorrectly combines terms and also changes the constant to 1. Option C treats 3x as 3x^2, although only one factor contains x in that product. Option D places incorrect coefficients and degrees in the regions. A useful check is that every part must come from one term on the first side and one term on the second side.