Visual models of identities

The direct answer is A: \(ab\) appears twice. To understand this visually, imagine a large square whose side is divided into lengths \(a\), \(b\), and \(c\) in both directions. The nine smaller regions include three squares, \(a^2\), \(b^2\), and \(c^2\), and six rectangles. The rectangle with sides \(a\) and \(b\) occurs in two positions: one has area \(ab\), and the other has area \(ba\). Since multiplication is commutative, \(ab=ba\), so together they contribute \(2ab\). Option A is correct. Option B is wrong because the \(a\)-by-\(a\) square occurs only once. Option C is wrong because the \(b\)-by-\(b\) square occurs only once. Option D is wrong because the \(c\)-by-\(c\) square also occurs only once. In fact, \((a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\). Memory cue: same-by-same gives one square; different-by-different gives a pair of equal rectangles.

The direct answer is B: there are three square terms. In an area model, divide each side of a large square into lengths \(a\), \(b\), and \(c\). A part of length \(a\) meeting another part of length \(a\) forms a square with area \(a^2\). Similarly, the \(b\)-by-\(b\) and \(c\)-by-\(c\) regions have areas \(b^2\) and \(c^2\). Thus the square terms are exactly \(a^2,b^2,c^2\), giving three terms. The full identity is \((a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\). Option A is wrong because it misses one of the three variables. Option B is correct because every variable creates one same-by-same square. Option C is wrong because there is no fourth variable or fourth diagonal square. Option D is wrong because six is the number of rectangular cross-term regions, not square regions. Memory cue: count the diagonal regions for square terms; the off-diagonal regions are mixed terms.

The direct answer is C: nine small sections. A square divided into three parts along one direction has three columns. Dividing it into three parts along the perpendicular direction gives three rows. Every row intersects every column, so the total number of small regions is rows multiplied by columns: \(3\times3=9\). In the identity model, these nine regions correspond to the products formed from \(a,b,c\) on the two sides: three diagonal square regions and six off-diagonal rectangular regions. Option A counts only one direction and therefore gives only three strips, not the complete two-dimensional grid. Option B is not the product of the row and column counts. Option C correctly counts all intersections. Option D has no basis in the stated division. Memory cue: for a rectangular grid, use number of rows times number of columns, not addition.

The direct answer is C: a square with side \(a+b\). Start with the identity \((a+b)^2=a^2+2ab+b^2\). A square whose side is split into lengths \(a\) and \(b\) in both directions contains one \(a^2\) square, one \(b^2\) square, and two rectangles of area \(ab\). Adding these regions gives \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Therefore option C matches the complete square exactly. Option A represents \((a-b)^2=a^2-2ab+b^2\), so its middle term has the wrong sign. Option B shows only \(a^2+b^2\) and misses the two cross-regions. Option D represents a product \((a+b)(a-b)=a^2-b^2\), not the given expression. Memory cue: plus \(2ab\) means the sum square; minus \(2ab\) means the difference square.

The direct answer is B: the square with side \(a-b\). The relevant identity is \((a-b)^2=a^2-2ab+b^2\). In a visual interpretation, begin with the large \(a\)-by-\(a\) square, remove two rectangular regions each having area \(ab\), and retain the small \(b\)-by-\(b\) square. The resulting signed area is \(a^2-ab-ab+b^2=a^2-2ab+b^2\). Option A gives \((a+b)^2=a^2+2ab+b^2\), so its middle term is positive. Option C gives \((ab)^2=a^2b^2\), which is not the same expression. Option D gives \((a+b^2)^2", which contains terms such as \(2ab^2\) and \(b^4\). Thus only B has the correct structure and sign. Memory cue: the negative middle term identifies the difference square.

The direct answer is B: 60. Use the identity \((a+b)^2=a^2+2ab+b^2\). Here the side 13 is split as \(10+3\), so \(a=10\) and \(b=3\). The middle part of the area model consists of two rectangles: one has area \(10\times3=30\), and the other has area \(3\times10=30\). Their combined area is \(2ab=2\times10\times3=60\). Option A is only the area of one rectangle, not both. Option B is correct. Option C does not follow from the identity and uses an incorrect product. Option D is \(10^2", the area of the large square portion, not the mixed part. As a check, \(13^2=100+60+9=169\). Memory cue: in a sum square, the middle term is twice the product of the two addends.

The direct answer is B: 25. Write 15 as \(10+5\). In the area model for \((10+5)^2\), the small square is formed by the second part of the side, which has length 5 in both directions. Therefore its area is \(5\times5=5^2=25\). The complete model has a large square of area \(10^2=100\), two rectangles each of area \(10\times5=50\), and the small square of area 25. These add to \(100+50+50+25=225=15^2\). Option A is only the side length 10, not an area. Option B is correct. Option C is the area of one mixed rectangle, not the small square. Option D is the area of the large square. Memory cue: the small corner square uses the second addend on both sides, so its area is the square of that addend.

The direct answer is A: \(2\times20\times2\). Use the difference-square identity \((a-b)^2=a^2-2ab+b^2\). For \((20-2)^2\), take \(a=20\) and \(b=2\). The expansion is \(20^2-2\times20\times2+2^2\). Therefore the mixed middle term, \(2ab=2\times20\times2=80\), is the term subtracted. Option A is correct. Option B, \(20^2\), is the first large square and is added. Option C, \(2^2\), is the small square and is also added. Option D is the final square being calculated, not a component that is subtracted. The numerical check is \(400-80+4=324=18^2\). Memory cue: for \((a-b)^2\), only the middle mixed term carries the minus sign; the two square terms remain positive.

The direct answer is B: \((x+3)(x+6)\). Expand it using the area model: \(x\cdot x=x^2\), \(x\cdot6=6x\), \(3\cdot x=3x\), and \(3\cdot6=18\). Adding the four regions gives \(x^2+6x+3x+18=x^2+9x+18\). Thus B produces exactly the required trinomial. A gives \(x^2+7x+2x+14=x^2+9x+14\), so its constant term is wrong. C gives a much larger middle coefficient and constant term. D gives \(x^2+18x+x+18=x^2+19x+18\), so its middle coefficient is wrong. A useful check for \((x+p)(x+q)\) is that \(p+q\) must equal 9 and \(pq\) must equal 18. Only 3 and 6 satisfy both conditions. Memory cue: the side numbers add to the coefficient of x and multiply to the constant term.

The direct answer is B: both expressions represent the same area. The area of a rectangle is length multiplied by breadth. Here both dimensions are \(x+4\), so its area is \((x+4)(x+4)\). When the same factor is multiplied by itself, it is written as a square: \((x+4)(x+4)=(x+4)^2\). Geometrically, a rectangle whose length and breadth are equal is a square, so the two models describe the same region. Option A is wrong because the two expressions are equal, not different. Option B is correct. Option C is wrong because a product of two lengths gives area, not perimeter; the perimeter would be \(4(x+4)\) for the square. Option D is wrong because \((x+4)^2\) is an area expression, not a line segment. Memory cue: a repeated factor \(u\cdot u\) is written \(u^2\), and equal rectangle dimensions make a square.

The correct answer is A: remove the square of side b from the square of side a. A square with side a has area a², and a square with side b has area b². When the smaller b-square is removed from the larger a-square, the remaining area is a² − b². This is exactly what a visual area model should show. Option B gives 2a², not a² − b². Option C gives 4b², so it does not represent the required subtraction. Option D shows only a length, whereas the expression represents an area. Do not confuse the difference of squares, a² − b², with the square of a difference, (a − b)². Memory cue: “large square minus small square.”

The correct answer is B. The area of a rectangle is length multiplied by breadth. With side lengths a + b and a − b, its area is (a + b)(a − b). Multiplying or using the difference-of-squares identity gives (a + b)(a − b) = a² − b². Thus the rectangle visually represents a² − b². Option A is the square of a single binomial and would require both sides to be a + b. Option C would require both sides to be a − b. Option D incorrectly omits the product term and is not generally equal to the rectangle’s area. Memory cue: conjugate factors, plus and minus the same term, produce a difference of squares.

The correct answer is C. Start with the area formula for a square: area = side × side = side². The identity (x + y)² = x² + 2xy + y² shows that the given expression is the square of x + y. In a visual model, one x-by-x square contributes x², one y-by-y square contributes y², and two x-by-y rectangles contribute xy + xy = 2xy. Together they make a square whose side is x + y. Option A gives (x − y)² = x² − 2xy + y², with a negative middle term. Option B is a product, not a side of this square. Option D is an expression for an area, not the required side. Memory cue: positive 2xy means “plus” between x and y.

The correct answer is D: x − 3. Begin with the square identity (x − a)² = x² − 2ax + a². The last term is 9, and 9 = 3², so a is 3. The middle term should then be −2(x)(3) = −6x, exactly as given. Therefore x² − 6x + 9 = (x − 3)², which represents the area of a square whose side is x − 3. A gives (x + 3)² = x² + 6x + 9, so its middle sign is wrong. B gives x² − 12x + 36, so both the middle coefficient and constant differ. C gives x² + 18x + 81, which does not match. Memory cue: a negative middle term in a perfect-square trinomial means subtraction in the side.

The correct answer is A. The numbers 21 and 19 are equally far from 20: 21 = 20 + 1 and 19 = 20 − 1. Therefore their product has the form (a + b)(a − b). The identity gives 21 × 19 = 20² − 1² = 400 − 1 = 399. Option B is used when the same binomial is multiplied by itself, not when conjugate binomials are multiplied. Option C also squares one binomial and has the wrong structure. Option D is the distributive property, but it is not the special identity being highlighted here. A useful memory cue is: numbers equally spaced around a middle number multiply as middle² minus distance².

The correct answer is C. The corner shape is a square whose side length is 7. The area of any square is side × side, so its area is 7 × 7 = 7² = 49 square units. The full square has side x + 7, but that does not change the area of this particular small corner square. Option A gives only the side length, not the area. Option B doubles the side and has no area meaning here. Option D incorrectly combines the variable x with 49; the corner square has side 7 alone. In an area model, each part must be calculated from its own two dimensions. Memory cue: the small constant square contributes the constant term 7² = 49.

The correct answer is B. The whole area is found by adding every separate region in the model: x² + 5x + 5x + 25. The two like terms combine as 5x + 5x = 10x, so the result is x² + 10x + 25. This also equals (x + 5)², which confirms the area-model identity. Option A leaves out one of the two 5x rectangles. Option C gives the wrong coefficient to x and changes the constant arrangement. Option D uses 5 instead of the required 25 for the small square. The key step is to count both matching rectangular parts before combining like terms. Memory cue: in (x + 5)², the middle contribution is 2 × x × 5 = 10x.

The correct answer is C. A square with side x + 2 can be divided into an x-by-x square, two rectangles each with area 2x, and a 2-by-2 square with area 4. Adding the parts gives x² + 2x + 2x + 4 = x² + 4x + 4. This is the identity (x + 2)² = x² + 2(x)(2) + 2². Option A omits both rectangular contributions. Option B includes only one 2x rectangle, so its middle term is too small. Option D has the sign used for (x − 2)², not (x + 2)². Memory cue: squaring a sum creates two positive cross-terms, so the middle term is +4x.

The correct answer is A. Use the square-of-a-difference identity: (x − 4)² = x² − 2(x)(4) + 4². Calculating the terms gives x² − 8x + 16. In a visual model, the x² part is the large square, the two cross-parts together contribute −8x in the algebraic rearrangement, and the small square contributes 16. Option B is the expansion of (x + 4)², so its middle sign is wrong. Option C uses only one x-by-4 contribution instead of two. Option D is the difference of squares x² − 4², which is a different expression from (x − 4)². Memory cue: the square of a difference has a negative middle term but a positive final square.

The correct answer is B. In the area model for (50 + 2)², the large square has side 50, so its area is 50². Calculate 50 × 50 = 2500. The other parts are two rectangles of area 50 × 2 = 100 each and a small square of area 2² = 4; together they eventually give 2704 for 52². However, the question asks only for the large 50² part, so the answer is 2500. Option A is the area of one 50-by-2 rectangle, not the large square. Option C is 50² + 100, and option D is the complete value of 52². Memory cue: identify the requested region before adding all regions.

The correct answer is C, \(1^2\). Begin with the identity \((a-b)^2=a^2-2ab+b^2\). Here, \(a=50\) and \(b=1\). In the visual square model, the large square has side 50, while the part removed from each side has width 1. The small corner formed by the removed strips is a square whose side is 1, so its area is \(1\times1=1^2\). Option A, \(50^2\), is the area of the original large square. Option B, \(49^2\), is the area of the final square whose side is 49, not the small corner square. Option D, \(2\times50\times1\), represents the combined area of two rectangular strips in the model, not the small square. Memory cue: in \((a-b)^2\), the small square has area \(b^2\), while the two strips together have area \(2ab\).

The correct answer is C, \(x^2+6x+9\). Treat \((x+3)(x+3)\) as a square with both side lengths split into \(x\) and 3. The area model then has four parts: an \(x\) by \(x\) square with area \(x^2\), two rectangles each measuring \(x\) by 3 with areas \(3x\) and \(3x\), and a 3 by 3 square with area 9. Adding them gives \(x^2+3x+3x+9=x^2+6x+9\). Option A includes only one \(3x\) rectangle. Option B incorrectly changes the two middle parts into \(9x\). Option D has the wrong constant term because the small square has area 9, not 3. The identity check is \((a+b)^2=a^2+2ab+b^2\), with \(a=x\) and \(b=3\).

The correct answer is A, the area model. An algebraic identity can be understood geometrically by representing expressions as lengths and products as areas. In an area model, a large rectangle or square is divided into smaller rectangles and squares. The area of the whole figure equals the sum of the areas of all its parts. For example, a square with side \(a+b\) can be divided into areas \(a^2\), \(ab\), \(ab\), and \(b^2\), giving \((a+b)^2=a^2+2ab+b^2\). Option B, the number-line model, mainly represents numbers and their positions or movements, not a sum of rectangular areas. Option C, the clock model, represents time and angles. Option D may explain an idea verbally, but it does not specifically use areas. Memory cue: if the diagram is divided into squares or rectangles and their areas are added, it is an area model.

The correct answer is A. In the square model for \((a+b)^2\), the side is formed by joining lengths \(a\) and \(b\). The complete square can be divided into one \(a^2\) square, two \(ab\) rectangles, and one \(b^2\) square, so the parts are added: \(a^2+2ab+b^2\). In the model for \((a-b)^2\), a length \(b\) is removed from a length \(a\), so some strips or regions are subtracted, producing \(a^2-2ab+b^2\). Option B is false because both models represent positive areas when the side lengths are valid. Option C is false because both are area models, not just lines. Option D reverses the signs: \((a-b)^2\) has side \(a-b\), not always \(a+b\). The sign inside the bracket is the key visual clue.

The correct answer is B, \(p+q\). Start with the standard identity \((p+q)^2=p^2+2pq+q^2\). The given total area is therefore the square of \(p+q\). Since the area of a square is \((\text{side})^2\), its side must be \(p+q\) (taking the usual non-negative length). Option A is not correct because \((p-q)^2=p^2-2pq+q^2\); its middle term has a minus sign. Option C, \(2pq\), is only the combined area of the two cross-rectangles in the visual model, not the side. Option D is a sum of two square areas and is not the square root of the complete expression. A useful check is to imagine four pieces: \(p^2\), two pieces of area \(pq\), and \(q^2\). Their outer side is obtained by adding the component lengths, giving \(p+q\).