Visual models of identities

The direct answer is A: 10. Multiply the two side lengths using the distributive property: (x+7)(x+3) = x·x + x·3 + 7·x + 7·3 = x² + 3x + 7x + 21. Combine like terms to obtain x² + 10x + 21. In the visual model, the x by x region has area x², the two linear rectangles have areas 3x and 7x, and the corner has area 21. Therefore the middle term is 10x and its coefficient is 10. B is the constant corner area 7×3, not the middle coefficient. C is not produced by the expansion. D counts only the 7x strip and ignores the 3x strip. A useful rule is (x+a)(x+b)=x²+(a+b)x+ab: add the side numbers for the middle coefficient and multiply them for the constant term.

The direct answer is A: the rectangle has sides r−s and r+s. The original square area is r² and the removed square area is s², so the remaining area is r²−s². The difference-of-squares identity says r²−s²=(r−s)(r+s). Cutting and rearranging does not change area; it only changes the shape of the L-shaped remainder. Therefore the rearranged rectangle must have one side r−s and the other side r+s. B would give area (r−s)², which is a different expression. C would give (r+s)² and would be larger than the original square in the usual positive-length setting. D gives area rs, not r²−s². Memory cue: difference of two squares becomes a rectangle whose sides are the difference and the sum.

The direct answer is A: x²+18xy+81y². The complete square is assembled from three kinds of regions: the main x by x square has area x², the two equal rectangles together have the already-given combined area 18xy, and the corner square contributes 81y². Adding the areas gives x²+18xy+81y². This is consistent with (x+9y)², because 81y²=(9y)² and 2·x·9y=18xy. B incorrectly uses only half of the combined rectangle area, as though one of the two rectangles had been omitted. C has a negative middle term, which would correspond to removing strips rather than adding them. D changes the corner area from 81y² to 9y², which is not what the model states. Memory cue: when the combined rectangle area is supplied, use it once as the full middle term.

The direct answer is B: 2ab+2bc+2ca. The full square has side a+b+c, so its area is (a+b+c)². The three diagonal square regions contribute a², b², and c². For each pair of different lengths, there are two equal rectangles: two a by b rectangles contribute 2ab, two b by c rectangles contribute 2bc, and two c by a rectangles contribute 2ca. Therefore the total rectangular area is 2ab+2bc+2ca. A counts only one rectangle for each pair and misses the matching partner. C is a length, not the total area. D counts only the three square regions and leaves out every rectangle. Memory cue: expanding a trinomial square creates each pairwise product twice.

The direct answer is option B: x² − 8x + 16. Begin with the original area, x². Two strips are removed, and their areas are 4x and 4x, so the total removed area is 4x + 4x = 8x. Therefore we subtract 8x from x². The two strips overlap at one corner; that common corner has been subtracted twice, so it must be added back once. Its area is 16. Hence the resulting area is x² − 8x + 16. This is also the identity (x − 4)² = x² − 8x + 16, because each removed strip has width 4. Option A has the wrong sign for the removed strips. Option C forgets the two strips and adds no correct-back corner. Option D only adds 16 and ignores both removals. Memory cue: remove means subtract, and a doubly counted corner is added back once.

The direct answer is option C: 2uv. Imagine the square of side u + v divided into parts of widths u and v in both directions. The u-by-u part has area u², and the v-by-v part has area v²; these are the differently colored squares mentioned in the question. The remaining two parts are rectangles. Each has length u and width v, so each area is uv. Adding the two equal rectangles gives uv + uv = 2uv. This is the middle term in the identity (u + v)² = u² + 2uv + v². Option A gives the area of only one rectangle, not both. Option B is a length expression, not the sum of the two rectangle areas. Option D is a difference of squares and does not represent these two positive rectangular regions. Memory cue: two identical u-by-v rectangles produce the cross-term 2uv.

The direct answer is option B: 4x² − 9. The area of a rectangle is length multiplied by width, so write (2x + 3)(2x − 3). This has the form (p + q)(p − q), whose identity is p² − q². Here p = 2x and q = 3. Therefore the area is (2x)² − 3² = 4x² − 9. If expanded directly, the middle terms are −6x and +6x, and they cancel: 4x² + 6x − 6x − 9 = 4x² − 9. Option A is the expansion of (2x + 3)², not a product of a sum and a difference. Option C incorrectly changes the leading coefficient. Option D corresponds to (2x − 3)², not the stated rectangle. Memory cue: conjugate factors (p+q)(p−q) leave a difference of squares and no middle term.

The direct answer is option A: 3a + 5b. The central square has area 9a², so its side is √(9a²) = 3a, taking the usual nonnegative lengths. The small square has area 25b², so its side is √(25b²) = 5b. In the tile arrangement, these side lengths join along the outer side of the complete square, so the whole side is 3a + 5b. The two equal strips provide a useful check: their total area is 2(3a)(5b) = 30ab, exactly as given. Option B uses the square areas as though they were side lengths. Option C uses subtraction, which would describe a difference arrangement rather than the stated joined tiles. Option D reverses the coefficients and does not give the sides of the two stated squares in the required order. Memory cue: take square roots of the corner-square areas, then add the adjacent side lengths.

The direct answer is option A: 49. The total area of the two equal strips is 14x, so each strip has area 7x. Since each strip is attached to a side of the x² square, its other dimension is x; therefore its width is (7x) ÷ x = 7. To complete the larger square, the missing corner is a square of side 7. Its area is 7² = 49. This is the completing-the-square pattern x² + 14x + 49 = (x + 7)². Option B is only the total strip area and is not a corner area. Option C is a length or algebraic strip area expression, not the constant corner area. Option D squares 14 instead of halving the middle coefficient. Memory cue: for x² + bx, halve b and square the result; here 14 ÷ 2 = 7 and 7² = 49.

The direct answer is option A. Start with a square of side y, whose area is y². The term −6y must come from removing two equal rectangular strips. If each strip has area 3y, their combined removed area is 3y + 3y = 6y. After removing them, the corner where the strips meet has been removed twice, so a small square of area 9 must be added back once. The resulting expression is y² − 6y + 9, which is (y − 3)². Option B adds the strips, producing positive middle terms instead of −6y. Option C changes only the constant part and cannot create the −6y term. Option D merely adds two separate squares and has no two-strip interaction. Memory cue: a negative middle term means strips are removed; the positive final constant is the restored overlap square.

The direct answer is option A: 4x and 9x. In a rectangle model, the two side strips must have coefficients whose sum gives the coefficient of x, 13, and whose product gives the constant term, 36. The numbers 4 and 9 satisfy both checks: 4 + 9 = 13 and 4 × 9 = 36. Thus the factorisation is x² + 13x + 36 = (x + 4)(x + 9), and the corresponding linear strips are 4x and 9x. Option B has sum 12 and product 36, so its middle coefficient is wrong. Option C has sum 13 but product 30, so its constant term is wrong. Option D has sum 13 but product 22, also incorrect. Memory cue: for x² + bx + c, find two numbers with sum b and product c; check both conditions, not just one.

The direct answer is option C: a − b. The area of the rearranged rectangle does not change, so area = length × width. We are given area a² − b² and length a + b. Therefore width = (a² − b²)/(a + b). Use the difference-of-squares identity a² − b² = (a + b)(a − b). Cancelling the nonzero factor a + b leaves width a − b. Equivalently, the tile pieces can be rearranged to show the factorisation directly. Option A would make the area (a+b)², not a²−b². Option B has the opposite order and would usually give a negative width when a>b. Option D is a product rather than the required remaining factor. Memory cue: difference of squares splits into sum times difference: a²−b²=(a+b)(a−b).

The direct answer is option B: 4ab. Expand both square areas carefully: (a + b)² = a² + 2ab + b², while (a − b)² = a² − 2ab + b². Subtract the second from the first: (a² + 2ab + b²) − (a² − 2ab + b²). The a² terms cancel and the b² terms cancel. The remaining terms are 2ab + 2ab = 4ab. Thus the visual difference between the two square areas is 4ab. Option A is only one cross-term, not the difference of both cross-terms. Option C is the sum of the square terms and does not result after subtraction. Option D is the difference a²−b², which comes from (a+b)(a−b), not from subtracting these two squares. Memory cue: the middle terms change from +2ab to −2ab, so their separation is 4ab.

The direct answer is option A: 4x², 10x, 10x, and 25. Split each side of the large square into lengths 2x and 5. The corner square with side 2x has area (2x)(2x) = 4x². Each of the two remaining rectangles has dimensions 2x and 5, so each area is (2x)(5) = 10x. The small corner square with side 5 has area 5 × 5 = 25. Adding the four pieces gives 4x² + 10x + 10x + 25 = 4x² + 20x + 25, which is exactly (2x + 5)². Option B incorrectly uses 2x² for the first square and 5x for the rectangles. Option C doubles each rectangle area incorrectly. Option D gives one incorrect rectangle of area 25x. Memory cue: square each side for corner squares, and multiply the two different side lengths for each rectangle.

The correct answer is B: the total area is \(a^2+2ab+b^2\), equal to \((a+b)^2\). A square with side \(a+b\) has area \((a+b)(a+b)\). The perpendicular cuts divide its sides into lengths \(a\) and \(b\), producing four regions: an \(a\)-by-\(a\) square with area \(a^2\), two \(a\)-by-\(b\) rectangles with areas \(ab\) each, and a \(b\)-by-\(b\) square with area \(b^2\). Their sum is \(a^2+ab+ab+b^2=a^2+2ab+b^2\). A lists only the two corner squares and leaves out both rectangles. C represents a difference of squares and would involve subtraction, not a complete partition into positive areas. D is a perimeter-type expression, not an area expression. The slash in the options is understood as separating the bilingual form, not as division. Memory cue: area of the whole equals the sum of every non-overlapping part.

The correct answer is B: the two \(xy\) rectangles still exist. The identity is \((x+y)^2=x^2+2xy+y^2\), and it is true for all values of \(x\) and \(y\) for which the algebraic expression is defined; it does not depend on how large the drawing looks. In the area model, the large \(x\)-by-\(x\) square has area \(x^2\), two rectangles each have area \(xy\), and the small \(y\)-by-\(y\) square has area \(y^2\). If \(y\) is small, the rectangles and the small square become narrow or small, but they do not vanish mathematically unless \(y=0\) exactly. A is wrong because size of the picture does not control truth of an identity. C is wrong because a small area is not automatically zero. D wrongly removes one \(xy\) rectangle and the \(y^2\) part. Memory cue: drawing scale may change appearance, but algebraic structure remains unchanged.

The correct answer is C: the side is \(4p-5q\). Use the identity \((u-v)^2=u^2-2uv+v^2\). Here \(16p^2=(4p)^2\) and \(25q^2=(5q)^2\). The middle term expected from \((4p-5q)^2\) is \(-2(4p)(5q)=-40pq\), exactly as given. Therefore \((4p-5q)^2=16p^2-40pq+25q^2\). A has a plus sign and would produce a positive middle term \(+40pq\), so it is not correct. B does not use the required square roots and its expansion is different. D is twice the correct side, so its square would be four times the given area. The negative middle term is the decisive clue: it means the two component lengths are subtracted in the side. Memory cue: square-root the first and last terms, then check the middle term and its sign.

In \((3x-2y)^2=(3x-2y)(3x-2y)\), the two removed corner rectangles each have area \((3x)(-2y)=-6xy\). There are two such opposite rectangles, so combined removed area is \(-6xy+(-6xy)=-12xy\).

The correct answer is A. Begin with a large square of side \(a\), whose area is \(a^2\), and remove a smaller square of side \(b\), whose area is \(b^2\). The remaining L-shaped region has area \(a^2-b^2\). By cutting and rearranging that remainder without changing its area, it can form a rectangle whose side lengths are \(a+b\) and \(a-b\). Its area is therefore \((a+b)(a-b)\), proving the identity visually. B does not identify the two square areas or the required rearrangement. C adds \(a^2\) and \(b^2\), whereas the identity requires their difference. D starts with the wrong square relationship and does not produce the factors \(a+b\) and \(a-b\). The important principle is conservation of area: cutting and rearranging changes shape, not total area. Memory cue: difference of squares becomes a rectangle after cut-and-rearrange.

The correct answer is B: the corner area is 64. The identity \(x^2+2xk+k^2=(x+k)^2\) contains two equal linear strips. Each strip has area \(xk\), so their combined area is \(2xk\). The question says this total is \(16x\), giving \(2xk=16x\). For nonzero \(x\), divide by \(2x\): \(k=8\). The corner is a square with side \(k\), so its area is \(k^2=8^2=64\). A is just the given coefficient value and is not the corner square. C is the value of k, not its area. D is \(16^2\), which incorrectly squares the total strip coefficient instead of finding k first. The visual model requires distinguishing a strip area \(xk\) from the corner area \(k^2\). Memory cue: first halve the total linear-strip area relative to x to find k, then square k for the corner.

The correct answer is A: the possible side lengths are (x - 5) and (x + 3). In a rectangle model, multiplying the two side expressions must give the area expression x^2 - 2x - 15. Multiply the proposed factors: (x - 5)(x + 3) = x^2 + 3x - 5x - 15 = x^2 - 2x - 15. Thus every term matches exactly. Another way to find the factors is to look for two numbers whose product is -15 and whose sum is -2. The numbers -5 and +3 satisfy both conditions. Option B gives x^2 + 8x + 15, so its middle and constant terms are wrong. Option C gives x^2 - 8x + 15: the middle term is wrong and the constant term has the wrong sign. Option D gives x^2 - 14x - 15, so its middle coefficient is also incorrect. Memory cue: for x^2 + bx + c, check both the product c and the sum b before accepting a factor pair.

The correct answer is B: the combined area is 4ab. Split each side of the square, whose total length is 2a + b, into parts 2a and b. The visual model then contains one square of side 2a, one square of side b, and two equal rectangles. Each rectangle has length 2a and width b, so the area of one rectangle is (2a)(b) = 2ab. Since there are two such rectangles, their combined area is 2ab + 2ab = 4ab. Option A is only the area of one of the two rectangles, not their total. Options C and D contain square-area terms such as a^2 or b^2, but the question asks only for the two mixed rectangular regions. The key point is that the side segment is 2a, not a. Memory cue: two identical rectangles contribute 2 × (length × width); here that is 2 × (2a × b) = 4ab.

The correct answer is A: the side of the square is (5x + 2y). Start by recognizing the perfect-square pattern a^2 + 2ab + b^2. Here, 25x^2 = (5x)^2 and 4y^2 = (2y)^2. The middle term should therefore be 2(5x)(2y) = 20xy, which is exactly the middle term given. Hence 25x^2 + 20xy + 4y^2 = (5x + 2y)^2, so the square’s side is 5x + 2y. Option B incorrectly treats the entire area expression’s coefficients as side terms. Option C is false because the expression is a perfect square. Option D would produce (5x - 2y)^2 = 25x^2 - 20xy + 4y^2, with a negative middle term, so it does not match. Memory cue: the sign of the middle term tells whether the two parts of the side are added or subtracted.

The correct answer is A: remove a square with side 7. The expression x^2 - 49 can be written as x^2 - 7^2 because 49 = 7^2. In an area model, x^2 represents a large square whose side is x, while 49 represents a smaller square with side 7. Removing that smaller square leaves the difference of two square areas, which can be cut and rearranged into a rectangle. This is the visual meaning of the identity x^2 - 7^2 = (x - 7)(x + 7). Option B is wrong because a square with side 49 has area 49^2, not 49. Option C has area (x + 7)^2 and is not the required constant piece. Option D adds an area and changes subtraction into a different expression. Memory cue: when a constant is subtracted from a square term, first write the constant as a square; here 49 becomes 7^2.

The correct answer is B: 16x. The larger square has area (x + 4)^2 and the inner square has area (x - 4)^2, so their difference is (x + 4)^2 - (x - 4)^2. Use the difference-of-squares identity A^2 - B^2 = (A - B)(A + B), taking A = x + 4 and B = x - 4. Then A - B = (x + 4) - (x - 4) = 8, and A + B = (x + 4) + (x - 4) = 2x. Therefore the difference is 8 × 2x = 16x. Option A uses only the first factor and misses the second. Option C is not the area difference; it is related only to part of an expansion. Option D is a constant and ignores the variable x. Memory cue: for two square areas, use difference of squares before expanding.