Visual models of identities

Direct answer: A is correct. The area of a square equals side × side. Here the side is a+b, so the area is (a+b)². To see the visual model, split the square into four parts: an a by a square with area a², a b by b square with area b², and two rectangles, each measuring a by b and each having area ab. Adding the four parts gives a² + ab + ab + b² = a² + 2ab + b². Option B represents a difference of two squares, not the area of a square whose side is a+b. Option C has only one ab rectangle and therefore misses one equal rectangular part. Option D is the perimeter, 2(a+b), not the area. Memory cue: for a plus b squared, remember square, two middle products, square: a² + 2ab + b².

Direct answer: B is correct. Add the area of every separate part. The square with side a has area a². The square with side b has area b². Each rectangle has dimensions a and b, so each has area ab. Since there are two equal rectangles, their combined area is 2ab. Therefore, total area = a² + ab + ab + b² = a² + 2ab + b². Option A counts only the two squares and leaves out both rectangles. Option C subtracts one rectangle, although all the pieces are included in the diagram and must be added; it also does not represent the stated arrangement. Option D doubles the square areas but ignores the rectangular areas. A visual-area question is solved by identifying each shape, calculating its area and then adding. Memory cue: two a-by-b rectangles always contribute 2ab.

Direct answer: C is correct. A square’s area is side². With side x+3, area = (x+3)². Apply the identity (p+q)² = p² + 2pq + q². Taking p=x and q=3 gives x² + 2(x)(3) + 3² = x² + 6x + 9. Option A has the wrong middle coefficient: it uses 3x instead of 6x, so it misses the two equal rectangular parts in the area model. Option B has the correct middle term but gives the constant term as 3 instead of 9; the small square has area 3²=9. Option D uses a negative middle term, which belongs to (x−3)², not (x+3)². Memory cue: in (x+3)², the middle term is twice x times 3, and the last term is 3 squared.

Direct answer: D is correct. Begin with the area of the large square: side a gives area a². The removed corner is a smaller square with side b, so its area is b². The remaining area is found by subtracting the removed part from the original part: a² − b². Option A adds the small square instead of removing it. Option B is the area of two rectangles with dimensions a and b in a different visual model; it is not the result of subtracting one square from another. Option C is the expansion associated with a square whose side is a+b, not a large square with a b-by-b corner removed. Option D correctly represents “large area minus removed area.” Memory cue: when a region is cut out, use subtraction; difference of squares is a²−b².

Direct answer: A is correct. The difference of squares identity is a²−b² = (a−b)(a+b). In the visual rearrangement, the two rectangles have side lengths or combined dimensions corresponding to a−b and a+b, so their total area is the product (a−b)(a+b). Expanding confirms it: (a−b)(a+b) = a² + ab − ab − b² = a²−b². Option B expands to a²+2ab+b², the square of a sum, so it has the wrong sign and structure. Option C expands to a²−2ab+b², the square of a difference, not a difference of two squares. Option D expands to a²+ab and does not equal a²−b² in general. Memory cue: difference of squares has one difference factor and one sum factor: “difference times sum.”

The direct answer is B: \(a^2-2ab+b^2\). The area of any square is side multiplied by itself, or (side)². Here the side is \(a-b\), so the area is \((a-b)^2=(a-b)(a-b)\). Multiply each term: \(a\cdot a=a^2\), \(a\cdot(-b)=-ab\), \((-b)\cdot a=-ab\), and \((-b)\cdot(-b)=b^2\). Adding these gives \(a^2-ab-ab+b^2=a^2-2ab+b^2\). A is wrong because \(a^2-b^2\) is the identity for \((a-b)(a+b)\), not the square of \(a-b\). C is the expansion of \((a+b)^2\), so its middle term has the wrong sign. D does not follow from multiplying the two binomials and is not a standard square identity. Memory cue: when a binomial with a minus sign is squared, the middle term is \(-2ab\), while the last term remains positive.

The direct answer is C: \((a-b)^2=a^2-2ab+b^2\). Begin with the large square of side \(a\), whose area is \(a^2\). Removing two rectangles, each measuring \(a\times b\), removes \(ab+ab=2ab\). The small square has side \(b\), so adding it contributes \(b^2\). Therefore the remaining area is \(a^2-2ab+b^2\). This remaining shape is a square with side \(a-b\), so its area is \((a-b)^2\). A has plus signs and describes \((a+b)^2\), where the rectangles are added rather than subtracted. B describes a product of a sum and a difference; its expanded form has no \(b^2\) term. D is an unrelated three-term expression and does not describe this area arrangement. Memory cue: two removed equal rectangles create the middle term \(-2ab\); the added small square supplies \(+b^2\).

The direct answer is D: \(x^2+7x+10\). The area of a rectangle is length multiplied by breadth, so calculate \((x+2)(x+5)\). A visual model divides the rectangle into four parts: an \(x\times x\) square with area \(x^2\), an \(x\times5\) rectangle with area \(5x\), a \(2\times x\) rectangle with area \(2x\), and a \(2\times5\) rectangle with area 10. Adding all parts gives \(x^2+5x+2x+10=x^2+7x+10\). A has the correct middle coefficient but the wrong constant term; 2 times 5 is 10, not 7. B incorrectly uses 10x as the middle term and also has the wrong constant. C uses only 3x for the two middle rectangles, although their total is 5x+2x=7x. D correctly includes every region of the model. Memory cue: for \((x+r)(x+s)\), the constant term is \(rs\) and the middle coefficient is \(r+s\).

The direct answer is A: \((x+4)^2\). The tiles represent the expression \(x^2+8x+16\). To recognize a perfect square, compare it with \((x+c)^2=x^2+2cx+c^2\). Since the constant term is 16, we take \(c=4\), because \(4^2=16\). Then the middle term should be \(2\cdot4\cdot x=8x\), exactly as shown by the eight \(x\)-tiles. Therefore \(x^2+8x+16=(x+4)^2\). B is wrong because \((x+8)^2=x^2+16x+64\), not the given expression. C would produce \(x^2-8x+16\), with a negative middle term. D leaves the expression ungrouped and omits the eight \(x\)-tiles. Memory cue: half the coefficient of \(x\) gives the added number; half of 8 is 4, and its square is 16.

The direct answer is B: \(x+5\). The small square has area 25. Since the area of a square is side squared, its side is \(\sqrt{25}=5\). In the usual visual model for \(x^2+2cx+c^2\), the two middle rectangles have dimensions \(x\times c\) and \(x\times c\), so their combined area is \(2xc\). Here, \(2x\cdot5=10x\), confirming that the added length is 5. Therefore the complete square has side \(x+5\). A would make the two middle rectangles total \(20x\), not \(10x\). C represents a subtraction and would lead to a negative middle term. D confuses the area 25 with the side length; the side is the square root of the area, not the area itself. Memory cue: in a perfect-square area model, the small square gives the extra side length after taking its square root.

The direct answer is C: two \(pq\) rectangles. Imagine a square whose side is \(p+q\). Divide both the horizontal and vertical sides into lengths \(p\) and \(q\). This creates four regions: a \(p\times p\) square of area \(p^2\), a \(q imes q\) square of area \(q^2\), and two cross-rectangles. Each cross-rectangle has dimensions \(p\) and \(q\), so each has area \(pq\). Together they contribute \(pq+pq=2pq\). Hence \((p+q)^2=p^2+2pq+q^2\). A is wrong because there are two cross-rectangles, one on each side of the diagonal arrangement. B is wrong because three such rectangles would give \(3pq\), which is not the square identity. D confuses the total number of regions, which is four, with the number of \(pq\) rectangles; the other two regions are squares. Memory cue: four regions, but only two are mixed \(p\)-by-\(q\) rectangles.

The direct answer is D: \(m^2-n^2\) is not one of the four individual parts. Split each side of the square into lengths \(m\) and \(n\). The partition creates one \(m imes m\) square with area \(m^2\), two rectangles measuring \(m imes n\), each with area \(mn\), and one \(n imes n\) square with area \(n^2\). Thus the complete area is \(m^2+mn+mn+n^2=m^2+2mn+n^2=(m+n)^2\). A, B, and C all describe actual regions in the partition; B occurs twice. D is a difference of two areas, not the area of one of these four regions. It belongs to the separate identity \(m^2-n^2=(m-n)(m+n)\). A common confusion is to treat any expression obtained from algebra as a physical part of the diagram. Memory cue: the four pieces of a sum-square model are two pure squares and two positive mixed rectangles.

The direct answer is A: \(a^2-b^2\). The area of a rectangle is length times breadth, so write \((a+b)(a-b)\). Multiply the terms: \(a(a-b)+b(a-b)=a^2-ab+ab-b^2\). The middle terms \(-ab\) and \(+ab\) cancel, leaving \(a^2-b^2\). This is the difference-of-squares identity \((a+b)(a-b)=a^2-b^2\). B is the expansion of \((a+b)^2\), which would be the area of a square whose two sides are both \(a+b\). C is the expansion of \((a-b)^2\), which would require both dimensions to be \(a-b\). D is related to adding lengths or perimeter-like expressions, not multiplying the rectangle's dimensions. Memory cue: a sum times the matching difference cancels the middle terms, so only the two squares remain.

The direct answer is B: 63. Notice that 7 is \(8-1\) and 9 is \(8+1\). Therefore \(7 imes9=(8-1)(8+1)\). Apply the identity \((u-v)(u+v)=u^2-v^2\): \(8^2-1^2=64-1=63\). Direct multiplication also confirms it: \(7 imes9=63\). A is wrong because it is two less than the correct value and does not follow from the identity. C is only \(8^2\); it forgets to subtract \(1^2\). D adds 1 instead of subtracting it from \(8^2\). The visual model shows two areas around the central square, and their product is one less than the square of the central number when the offsets are 1. Memory cue: numbers equally spaced around \(n\), namely \(n-1\) and \(n+1\), multiply to \(n^2-1\).

The direct answer is C: 144. First find the side: \(10+2=12\). The area of a square is side squared, so the area is \(12^2=144\). The visual model explains the same result by splitting the square into four parts: a \(10 imes10\) square with area 100, two \(10 imes2\) rectangles with areas 20 and 20, and a \(2 imes2\) square with area 4. Their total is \(100+20+20+4=144\). A is missing too much area and does not equal the expansion. B omits the small \(2 imes2\) square because it is only 100+20. D includes 100+20+4 but forgets the second middle rectangle. C includes all four regions and agrees with \((10+2)^2=10^2+2(10)(2)+2^2\). Memory cue: when squaring a sum, count both equal middle rectangles; that is the \(2ab\) term.

The direct answer is D, 100. Area tells us how much surface each part covers, so we add the areas of all four non-overlapping parts. The first square has area 6² = 36. Each rectangle has area length × breadth = 6 × 4 = 24, and there are two such rectangles, giving 24 + 24 = 48. The final square has area 4² = 16. Therefore, total area = 36 + 48 + 16 = 100. This also matches the identity (6 + 4)² = 6² + 2(6)(4) + 4² = 100. Option A, 76, is too small and does not include all four parts. Option B, 84, also results from an incomplete or incorrect addition. Option C, 96, is short by 4 and usually indicates that one part has been counted incorrectly. Option D, 100, includes both rectangles and both square regions, so it is correct. Memory cue: in a square model of (a+b)², remember square, two rectangles, square: a² + 2ab + b².

The direct answer is A, 9801. Begin with the difference-of-two-terms square identity: (a−b)² = a² − 2ab + b². Here a = 100 and b = 1 because 99 = 100−1. First calculate a² = 100² = 10,000. Next calculate the middle term 2ab = 2×100×1 = 200. Finally b² = 1² = 1. Substitution gives (100−1)² = 10,000 − 200 + 1 = 9,801. Option A is therefore correct. Option B, 9901, subtracts only 100 or otherwise mishandles the middle term; the correct middle term is 200. Option C, 9999, is close to 10,000 but does not apply the square identity correctly. Option D, 10001, is greater than 10,000, whereas squaring 99 must give a number slightly less than 100². A quick check is 99×99 = 9801. Memory cue: for a number one less than 100, subtract 200 and add 1.

The direct answer is B, (x+6)². Use the pattern (x+a)² = x² + 2ax + a². The constant term is 36, whose square root is 6, so the last part suggests a = 6. The middle term then becomes 2×x×6 = 12x, exactly the middle term in the expression. Therefore x² + 12x + 36 = (x+6)². In a visual model, x² is the large square, 6x and 6x are two equal rectangles, and 36 is the small 6-by-6 square. Option A, (x+12)², would expand to x² + 24x + 144, so both the middle and constant terms are wrong. Option B matches every term. Option C, (x−6)², expands to x² − 12x + 36; its middle term has the wrong sign. Option D, (x+3)², gives x² + 6x + 9, so its numbers are too small. Memory cue: take the square root of the constant and use half of the middle coefficient to confirm the sign.

The direct answer is C, (y+4)². Add the four regions because they form the whole square: y² + 4y + 4y + 16 = y² + 8y + 16. Now compare this with (y+a)² = y² + 2ay + a². Taking a = 4 gives y² + 2×4×y + 4² = y² + 8y + 16, so the model represents (y+4)². Option A, (y+8)², would have middle term 16y and constant 64, not 8y and 16. Option B, (y−4)², has middle term −8y, so its sign is wrong. Option C has exactly the required large square, two 4y rectangles, and 16 corner square. Option D, y²+8y+4, is missing 12 from the constant part; the corner square is 4² = 16, not 4. A useful memory cue is that two identical rectangles each measuring 4y combine to make 8y, while the corner square contributes 16.

The direct answer is C: the area of the small square whose side is s. In the model for a square with side r+s, each side is split into lengths r and s. This creates four regions: an r-by-r square with area r², two rectangles each with area rs, and an s-by-s square with area s². Since the two dimensions of the last region are both s, its area is s×s = s². Option A is incorrect because the two rectangles together have area 2rs, not s². Option B describes the large square with side r and area r². Option C correctly identifies the small corner square with side s. Option D is also incorrect because s² is an area, while perimeter is a length measurement and would involve adding side lengths. The important distinction is that a squared term usually represents the area of a square whose side has that symbol. Memory cue: same side multiplied by itself gives a square area: s×s = s².

The direct answer is B: x² and 4. Apply (a−b)² = a² − 2ab + b² with a=x and b=2. This gives (x−2)² = x² − 4x + 4. In the visual interpretation, the large x-by-x square contributes x², the two strips together contribute the negative middle term −4x, and the small 2-by-2 corner square contributes 4. Thus both square-area terms, x² and 4, are positive. Option A, −4x, is not positive; it is the negative middle contribution caused by subtracting the strips. Option B is correct because x² and 4 have positive signs. Option C, −x², incorrectly changes the sign of the large square’s area. Option D, −4, incorrectly changes the sign of the small corner square; 2² is +4. In ordinary geometric area, a square region has a nonnegative area. Memory cue: in (a−b)², the middle term is negative, but the first and last square terms remain positive.

The direct answer is A, (a+b)². Divide each side of a square into lengths a and b. The resulting model has one a-by-a square of area a², one b-by-b square of area b², and two equal rectangles. Each rectangle has dimensions a and b, so each has area ab. Adding the two rectangle areas gives ab+ab = 2ab. Therefore the complete area is a²+2ab+b² = (a+b)². Option A is correct because its middle term is positive 2ab. Option B, (a−b)², has the algebraic form a²−2ab+b²; its two mixed regions contribute a negative middle term in the identity. Option C contains only one ab, not two equal rectangle contributions. Option D, a²−b², is a difference of squares and does not contain a 2ab middle term. The model helps explain why the coefficient 2 appears: there are two, not one, identical mixed rectangles. Memory cue: plus-square means two positive ab rectangles; minus-square means a negative middle term.

The direct answer is B, 4 parts. A rectangle representing (x+1)(x+4) has one side split into two pieces, x and 1, and the other side split into two pieces, x and 4. A partition with two pieces in one direction and two in the other creates 2×2 = 4 smaller rectangles. Their areas are x×x = x², x×4 = 4x, 1×x = x, and 1×4 = 4. Together they explain the expansion x²+4x+x+4 = x²+5x+4. Option A, 2, counts only the pieces on one side and ignores the second split. Option B, 4, is correct because every piece from the first factor meets every piece from the second factor. Option C, 6, has no corresponding 2-by-2 grid structure. Option D, 8, doubles the required number and does not represent the simple rectangular partition. A useful method is to count factor parts, not algebraic terms: two parts times two parts gives four regions. Memory cue: for (two-term)(two-term), the basic grid has four cells.

The direct answer is C, 5x. Use the rectangle model or distribute each term: (x+1)(x+4) produces x·x = x², x·4 = 4x, 1·x = x, and 1·4 = 4. The two terms containing exactly one x are 4x and x. They are like terms because both have x to the first power, so their coefficients can be added: 4x+x = 5x. Thus the full expansion is x²+5x+4. Option A, 4x, is only the area of one mixed rectangle and leaves out the other x-term. Option B, x, is the other mixed rectangle alone. Option C correctly combines both mixed regions. Option D, 6x, adds the constants 4 and 1 to the coefficient incorrectly; constants are not x-terms. The important rule is to combine only like terms: 4x and x can combine, but x² and x cannot, and 4 cannot combine with them. Memory cue: in a two-by-two product grid, add the two off-diagonal rectangles to obtain the middle term.

The direct answer is D, 6. In the rectangle model, split the first side into a and 3, and the second side into a and 2. The region formed by the two constant pieces has dimensions 3 and 2, so its area is 3×2 = 6. Expanding confirms the result: (a+3)(a+2) = a²+2a+3a+6 = a²+5a+6. The final constant term is the number-only region. Option A, 3a, is the mixed region produced by 3×a, so it contains a and is not constant. Option B, 2a, is the other mixed region, a×2. Option C, 5a, is the combined total of the two a-terms, not one separate constant region. Option D, 6, is correct because it uses only the constant dimensions 3 and 2. A constant part contains no variable; that is the key identification rule. Memory cue: multiply the numbers at the ends of the factors to find the constant term: 3×2 = 6.