Visual models of identities

The direct answer is option A: p²+4p+4. The original p by p square has area p². Each added strip has width 2 and length p, so each contributes 2p. The two strips together contribute 2p+2p=4p. Where the strips meet, there is a corner square of side 2, whose area is 2×2=4. Thus the complete area is p²+2p+2p+4=p²+4p+4. This can also be checked by observing that the new outer side is p+2, so the area is (p+2)²=p²+4p+4. Option B includes only one strip, so it misses 2p. Option C includes the two strips but gives the corner area as 2 rather than 4. Option D uses subtraction signs, although every part is being added. The memory cue is: for a square enlarged by k, use (p+k)²=p²+2pk+k².

The direct answer is option C. Imagine a square whose side is m+7 and divide each side into lengths m and 7. The resulting large square contains one m by m square with area m², two rectangles with dimensions m by 7 and area 7m each, and one 7 by 7 corner square with area 49. Therefore the complete partition is m²+7m+7m+49=m²+14m+49, which is exactly (m+7)². Option A has only one 7m rectangle, so one strip is missing and it gives an incomplete area. Option B combines the two middle rectangles as 14m but then lists 49 twice, adding an extra corner; there is only one 7 by 7 corner. Option D uses incorrect dimensions and gives 49m instead of the required strip area 7m. A useful check is the identity (a+b)²=a²+2ab+b²: the middle contribution must be 2m×7=14m.

The direct answer is option D: −8x+32. Expand the square of a difference carefully: (x−4)²=x²−2(x)(4)+4²=x²−8x+16. The student’s expression x²−16 is missing the middle term −8x and also has the wrong constant term: it has −16 instead of +16. To find the complete correction, subtract the student’s result from the correct result: (x²−8x+16)−(x²−16)=−8x+32. Therefore adding −8x+32 to x²−16 produces x²−8x+16. Option A has the wrong sign and does not include the constant correction. Option B is only half of the middle-term magnitude. Option C has no basis in the expansion. Option D includes both required changes. The common confusion is between (x−4)²=x²−8x+16 and x²−16, which is the different identity x²−4² only when the expression is a difference of squares, not a square of a difference.

The direct answer is option B: x+2 and x+9. In an area model, the central x² tile comes from multiplying x by x. A strip of area 2x has dimensions x and 2, while a strip of area 9x has dimensions x and 9. The remaining corner must then have area 2×9=18. Therefore the two side lengths are x+2 and x+9. Multiplying confirms this: (x+2)(x+9)=x²+9x+2x+18=x²+11x+18, exactly the sum of the four model parts. Option A incorrectly uses 11, which is only the combined coefficient of the two x terms, as a side length; 18 is the corner area, not a side. Option C would give corner product 3×6=18, but its strips would be 3x and 6x, not 2x and 9x. Option D has negative side pieces and would produce negative middle terms. Always match each strip and check the corner product.

The direct answer is option A: 4x² and 25. Split the outer side 2x+5 into two lengths, 2x and 5. The main square has side 2x, so its area is (2x)²=4x². The small corner has side 5, so its area is 5²=25. For completeness, the two connecting rectangles each have area (2x)(5)=10x, giving the full expansion (2x+5)²=4x²+20x+25. Option B is wrong because (2x)² is 4x², not 2x², and 5 is a length rather than the corner’s area. Option C gives products related to sides but not the areas of the two square pieces. Option D gets the corner right but still miscalculates the main square. The key rule is that an area of a square is side multiplied by itself: coefficients and constants must both be squared.

The direct answer is option B: 6x. Divide each side of the large square into lengths 3x and 1. This creates a main square of side 3x with area (3x)²=9x², a small corner square of side 1 with area 1²=1, and two equal middle rectangles. Each middle rectangle has dimensions 3x by 1, so each area is 3x. Together their area is 3x+3x=6x. This is the middle term 2ab in the identity (a+b)²=a²+2ab+b², with a=3x and b=1. Option A is only the area of one rectangle, not both. Option C is the main square’s area. Option D is the corner square’s area. A useful memory cue is that the two middle pieces always contribute twice the product of the two component side lengths.

The direct answer is option A. Since 81=9², the expression z²−81 means the area of a square of side z minus the area of a square of side 9. Imagine cutting the smaller 9 by 9 square from the larger z by z square. The remaining L-shaped region can be rearranged into a rectangle whose side lengths are z−9 and z+9. Thus z²−81=(z−9)(z+9), the difference-of-squares identity. Option B forms (z+9)², which adds the middle and corner areas rather than subtracting 81. Option C forms (z−9)², which equals z²−18z+81 and is a different expression. Option D gives an area 9z, not z²−81. The visual memory cue is: subtracting two square areas produces two factors, the difference and the sum of their side lengths.

The direct answer is option C: 6x. A square expression can be viewed as a large square together with two equal rectangular strips and a corner square. Here x^2 is the area of the square with side x, and 36 is the area of the corner square with side 6. The two equal middle rectangles together have area 12x, so each one has area 12x ÷ 2 = 6x. Option A is the combined area of both rectangles, not one rectangle. Option B, 36x, incorrectly mixes the constant corner value with x; the corner area is 36. Option C correctly gives the area of one equal strip. Option D, x+6, is a side length in the model, not an area. The identity is (x+6)^2 = x^2+12x+36. Memory cue: the middle term is twice one equal rectangle, so divide its coefficient by two.

The direct answer is option B: 24. The area of the rectangle is (x+3)(x+8). In an area grid, multiply every part of the first side by every part of the second side: x·x gives x^2, x·8 gives 8x, 3·x gives 3x, and 3·8 gives 24. The small corner uses the two constant lengths, 3 and 8, so its area is their product, 3×8=24. Option A is 3+8=11, the sum, not the corner area. Option B is correct because a rectangle’s area is length times breadth. Option C is 8−3=5, the difference, which has no role here. Option D represents the two variable strips, 3x+8x=11x, so it is not constant. The complete expansion is x^2+11x+24. Memory cue: constant corner means constant times constant.

The direct answer is option C: a square with side (2a−3). Use the identity (p−q)^2=p^2−2pq+q^2. Taking p=2a and q=3 gives (2a−3)^2=(2a)^2−2(2a)(3)+3^2=4a^2−12a+9, exactly the given expression. Option A gives (2a+3)^2=4a^2+12a+9, so its middle term has the wrong sign. Option B gives 16a^2−72a+81; its first and last terms do not match. Option C matches the first term, middle term, and constant term. Option D gives 16a^2+24a+9, so both the leading and middle terms are wrong. In a visual model, the negative middle term means the strips are subtracted rather than added. Memory cue: take square roots of the first and last terms, then use the middle-term sign.

The direct answer is option A. Add the four region areas: a^2+ab+ab+b^2=a^2+2ab+b^2. The complete square has side length a+b, because one part of each side has length a and the other has length b. Therefore its total area is (a+b)^2, giving (a+b)^2=a^2+2ab+b^2. Option A is correct because there are two positive ab rectangles. Option B would require the two middle regions to be negative, as in a square of a difference. Option C describes a difference of two squares and uses multiplication of two binomials, not four positive regions of this model. Option D omits the two ab regions, so it misses part of the area. Memory cue: two identical ab rectangles combine to make 2ab.

The direct answer is option B: 8n. A square with side n+4 has area (n+4)^2. Using (p+q)^2=p^2+2pq+q^2 with p=n and q=4 gives n^2+2(n)(4)+4^2=n^2+8n+16. The given n^2 is the large variable square and 16 is the constant corner square. The missing part consists of two equal rectangular strips, each with area n×4=4n. Together they have area 4n+4n=8n. Option A gives only one strip. Option B correctly includes both strips. Option C incorrectly multiplies 16 by n and does not represent either strip. Option D is a side length, not an area. Memory cue: the middle term 2n·4 counts two identical strips.

The direct answer is option A: x^2+7x+10. Multiply each row label by each column label. The four cells are x·x=x^2, x·2=2x, 5·x=5x, and 5·2=10. Adding them gives x^2+2x+5x+10=x^2+7x+10. Option A includes every cell and combines the two middle terms correctly. Option B has the coefficients and constant in the wrong places; 10 is the product of the constants and must be the constant term. Option C includes only one variable strip and omits the other strip and the corner. Option D uses a negative middle term even though all row and column labels are positive. The model is the distributive property in picture form. Memory cue: fill all four cells before collecting like terms.

The direct answer is option A: (100−2)^2. Since 98=100−2, the square can be represented by the identity (a−b)^2=a^2−2ab+b^2. Thus 98^2=(100−2)^2=10000−400+4=9604. Option A is especially useful because 100 is a nearby round number, making the areas and products easy to calculate. Option B is mathematically possible because 98=90+8, but it is less convenient and does not use the nearest simple round number. Option C is not the correct representation of 98^2: 100^2−2^2 is a difference of squares, not (100−2)^2, and it equals 9996. Option D is the area of a 98 by 100 rectangle, not a 98 by 98 square. Memory cue: for a number near 100, write it as 100 plus or minus the small difference.

The direct answer is option B: 4x. Begin with an x by x square, whose area is x^2. To reduce both side lengths from x to x−2, the model removes two rectangular strips: one has area x·2=2x along one side, and the other has area x·2=2x along the other side. Therefore the total rectangular strip area removed is 2x+2x=4x. The small 2 by 2 overlap has area 4, but it is not a rectangular strip counted in this question; it is added back when completing the identity. Option A counts only one strip. Option B counts both strips. Option C is only the overlap square. Option D is not the removed area; it combines unrelated terms. The identity is (x−2)^2=x^2−4x+4. Memory cue: subtracting 2 from both dimensions creates two 2x strips.

The direct answer is option B: the sides are (x+3) and (x+7). To factor x^2+10x+21, find two numbers whose sum is 10 and whose product is 21. The numbers 3 and 7 satisfy both conditions: 3+7=10 and 3×7=21. Therefore (x+3)(x+7)=x^2+7x+3x+21=x^2+10x+21. Option A has constants 1 and 21, whose sum is 22, so its middle term is wrong even though the product is 21. Option B satisfies both the middle coefficient and the constant term. Option C has 2+8=10, but 2×8=16, not 21. Option D gives positive constant 21 but negative middle term −10x, so it represents a different expression. Memory cue: for x^2+bx+c, check both sum b and product c.

The direct answer is option A. The expression 9r^2−16 is a difference of two perfect squares. Since 9r^2=(3r)^2, the first area is a square with side 3r. Since 16=4^2, the second area is a square with side 4. Therefore the correct visual model removes the 4-by-4 square from the square whose side is 3r. Algebraically, 9r^2−16=(3r)^2−4^2=(3r+4)(3r−4). Option A correctly identifies both square roots and the subtraction. Option B mistakes each area for a side length: 9r and 16 are not the required side lengths. Option C makes a square of a sum and would produce a positive middle term 24r. Option D also uses incorrect side parts and would produce additional terms. Memory cue: in A^2−B^2, draw the square of side A and remove the square of side B.

Correct answer: B, \(10x\). In an area model, multiply the length of one side by the length of the corresponding part of the other side. Split the sides into \(x+1\) and \(x+9\). The four regions have areas \(x\cdot x=x^2\), \(x\cdot9=9x\), \(1\cdot x=x\), and \(1\cdot9=9\). The two middle strips are therefore \(9x\) and \(x\). Adding them gives \(9x+x=10x\). Option A, \(9x\), counts only one strip. Option B combines both middle strips and is correct. Option C is the large square-like term from multiplying \(x\) by \(x\), not a middle strip. Option D is the small corner area. Memory cue: the two cross-strips usually combine into the coefficient of the middle term.

Correct answer: A, \(a^2+2ab+b^2\). Each side of the large square has length \(a+b\), so its area is \((a+b)^2\). Dividing both directions creates four regions: an \(a\times a\) square of area \(a^2\), two rectangles of area \(ab\) each, and a \(b\times b\) square of area \(b^2\). Adding all regions gives \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Option A is exactly this result. Option B is a difference of squares, so it has no two positive middle rectangles. Option C is the identity for \((a-b)^2\), with negative middle terms. Option D incorrectly counts two copies of each square and omits the correct two \(ab\) contributions. Memory cue: for a plus square, the middle term is always \(+2ab\).

Correct answer: C, \((a+b)(a-b)\). The difference-of-squares identity is \(a^2-b^2=(a+b)(a-b)\). Visually, after the smaller \(b\times b\) square is removed from the \(a imes a\) square, the remaining L-shaped region can be cut and rearranged into a rectangle. The rearranged rectangle has one side \(a+b\) and the other side \(a-b\), so its area is \((a+b)(a-b)\). Option A adds the two square areas instead of subtracting one. Option B is only one length, not an area. Option C gives exactly the required product. Option D represents \((a-b)^2=a^2-2ab+b^2\), which is a different expression. Memory cue: difference of two squares becomes sum times difference.

Correct answer: A, \((a+b)^2=a^2+2ab+b^2\). The entire figure is a square whose side is \(a+b\), so its area is \((a+b)^2\). The four partitioned areas are \(a^2\), \(ab\), \(ab\), and \(b^2\). Their total is \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Option A includes both equal rectangular regions. Option B has a negative middle term and belongs to a side of \(a-b\). Option C incorrectly says the two corner squares alone equal the whole square; the two \(ab\) rectangles are missing. Option D includes only one of the two rectangles. A useful check is to count every region before simplifying: two \(ab\) regions must produce \(2ab\).

Correct answer: A, \((x+y)^2=x^2+2xy+y^2\). Add the four given areas: \(x^2+xy+xy+y^2=x^2+2xy+y^2\). The complete square has side length \(x+y\), so its total area is \((x+y)^2\). Option A matches both the outer square and the four inner regions. Option B has \(-2xy\), which would require subtractive strips and represents \((x-y)^2\). Option C represents a rectangle with sides \(x+y\) and \(x-y\), not this four-part positive square. Option D forgets the two rectangular areas and is therefore incomplete. Memory cue: two identical \(xy\) rectangles give \(2xy\); their areas are positive in this model.

Correct answer: A. Divide both sides of the square into lengths \(a\) and \(b\). The horizontal and vertical divisions create four regions: one \(a\times a\) square with area \(a^2\), one \(b\times b\) square with area \(b^2\), and two rectangles, each \(a\times b\), with area \(ab\). Thus the total is \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Option B has the wrong number of square regions and gives the wrong total. Option C misses one rectangle, so it gives only one \(ab\). Option D describes triangles rather than the natural rectangular partition; two such triangles would not be the two full cross-rectangles in this model. Memory cue: split both directions, then count 1 large-type square, 1 small-type square, and 2 cross-rectangles.

Correct answer: A, \(4u^2-4u+1\). The area of the square is \((2u-1)^2\). Use \((p-q)^2=p^2-2pq+q^2\), with \(p=2u\) and \(q=1\). Then \((2u)^2-2(2u)(1)+1^2=4u^2-4u+1\). In the visual model, the large square contributes \(4u^2\), two equal strips together contribute \(-4u\), and the small corner contributes \(+1\). Option B counts only a single \(-2u\) strip. Option C incorrectly squares the main term. Option D has the wrong positive sign and would match \((2u+1)^2\). Memory cue: the coefficient of the middle term is twice the product of the two parts, and a minus inside the bracket makes it negative.

Correct answer: B, \(2x\) and \(3x\). In a rectangle or algebra-tile model for \(x^2+5x+6\), the two middle strips must add to \(5x\), while their constant factors must multiply to 6. The numbers 2 and 3 satisfy both conditions: \(2+3=5\) and \(2\cdot3=6\). Therefore the strips are \(2x\) and \(3x\), and the factorisation is \((x+2)(x+3)\). Option A adds to \(11x\) and has the wrong constant product. Option C has coefficients 1 and 6, whose sum is 7, not 5. Option D has coefficients 1 and 5, whose product is 5, not 6. Memory cue: for \(x^2+bx+c\), find two numbers whose sum is \(b\) and product is \(c\).