Visual models of identities

The direct answer is A: 11a. The rectangle’s area is (a+2)(a+9). Divide it into four smaller rectangles: a×a=a^2, a×9=9a, 2×a=2a, and 2×9=18. The two regions containing one variable factor are the cross-regions. Their areas, 9a and 2a, are like terms, so they combine as 9a+2a=11a. Thus the full expansion is a^2+11a+18, and the middle term is 11a. B, 18, is the constant term formed by multiplying 2 and 9. C, 7a, comes from subtracting the coefficients instead of adding them. D has degree two and represents neither cross-region; the a×a region is a^2, not 2a^2. A reliable area-model method is to write all four products before combining like terms. The middle term comes from the two cross-products, not from the corner.

The direct answer is A: (a+b)^2=a^2+2ab+b^2. Each side of the outside square is split into lengths a and b, so the full side is a+b and the total area is (a+b)^2. The four smaller regions have areas a×a=a^2, a×b=ab, b×a=ab, and b×b=b^2. Adding them gives a^2+ab+ab+b^2=a^2+2ab+b^2. Therefore the model represents option A. B would require the two cross-regions to be negative, which is not the case when the lengths are added. C and D describe the difference-of-squares identity: they involve one plus factor and one minus factor, not a square divided into four positive regions with side a+b. The main memory cue is to count both ab rectangles; forgetting one changes 2ab into ab and leads to an incomplete expansion.

The direct answer is A: (a+b)^2=a^2+2ab+b^2. The model contains one a^2 square, two rectangles of area ab, and one b^2 square. Adding their areas gives a^2+ab+ab+b^2=a^2+2ab+b^2. Because the outer figure is a square made by joining lengths a and b along each side, its side length is a+b and its area is (a+b)^2. Hence the two descriptions of the same total area are equal, giving option A. B has negative cross-regions and represents (a-b)^2, not the stated positive areas. C and D are equivalent forms of the difference-of-squares identity and require factors a+b and a-b; they do not match this four-part plus-area model. Be careful not to count only one ab rectangle: the two cross-regions together are 2ab. This is why the coefficient of the middle term is 2.

The correct answer is A. The area is the product of the side lengths: \((q+1)(q+6)=q^2+6q+q+6=q^2+7q+6\). The two middle terms combine to make \(7q\).

The correct answer is B. The constant corner is a square whose side is 5, so its area is \(5\times5=25\). The terms involving \(p\) belong to the other regions of the area model.

The correct answer is A: he should take two 3x strips. Expand the square identity: (x+3)²=x²+2×x×3+3²=x²+6x+9. The visual model has one x² square, two rectangular strips of dimensions x by 3, and one 3 by 3 corner square. Each strip has area 3x, so together they contribute 6x. Taking only one strip gives x²+3x+9, which is incomplete and does not equal the square’s area. Option B is wrong because the middle contribution is 6x, not 9x. Options C and D remove parts that are required. Memory cue: squaring a binomial creates two cross-products, not one.

The correct answer is A. If the two parts a and b make one side of the large square, then the side length is a+b. The area of a square is side multiplied by itself, so the total area is (a+b)². Expanding gives (a+b)(a+b)=a²+ab+ab+b²=a²+2ab+b². The visual model has one a-by-a square with area a², one b-by-b square with area b², and two rectangles, each with area ab; together the rectangles contribute 2ab. Option B is the difference-of-squares expression and would require a product such as (a+b)(a−b). Option C leaves out the two rectangular regions. Option D is the perimeter-like expression 2a+2b, not an area. First identify the side, then square it. Memory cue: a square built from a and b contains two cross-rectangles, so the middle term is 2ab.

The correct answer is B. The side of the square is x+3, so its area is (x+3)². Apply (p+q)²=p²+2pq+q² with p=x and q=3: (x+3)²=x²+2(x)(3)+3²=x²+6x+9. In the visual model, x² is the large x-by-x square, 3²=9 is the small corner square, and the two equal rectangles together have area 3x+3x=6x. Option A forgets one of the two equal rectangles, so it has 3x instead of 6x. Option C places the coefficients incorrectly and is not produced by the identity. Option D has a negative middle term, which would correspond to (x−3)², not (x+3)². A plus sign inside the bracket produces a plus middle term. Memory cue: square the first term, double the product, then square the second term.

The correct answer is C. Removing a strip of width b from a square whose side is a leaves a smaller square with side a−b. The area of that remaining square is therefore (a−b)². Expanding it gives (a−b)(a−b)=a²−ab−ab+b²=a²−2ab+b². The two negative ab terms represent the two rectangular portions removed in the area model, while the b-by-b corner is counted back, producing +b². Option A describes adding a strip, not removing one. Option B is not a valid equality in general; the difference of squares is a²−b²=(a+b)(a−b). Option D is not the square identity. The key is to connect removal with subtraction in the side length. Memory cue: a minus inside a square creates a negative double-product term, but the last square remains positive.

The correct answer is D: a²−b². The area of a rectangle is the product of its length and breadth, so the area here is (a+b)(a−b). Multiply term by term: a·a=a², a·(−b)=−ab, b·a=+ab, and b·(−b)=−b². Combining these gives a²−ab+ab−b². The middle terms cancel because −ab+ab=0, leaving a²−b². Option A is (a+b)², which would be the area of a square with both sides a+b. Option B has an extra, incorrect ab term. Option C loses the negative sign before b² and therefore does not represent this product. Option D is exactly the difference-of-squares identity (a+b)(a−b)=a²−b². Memory cue: sum multiplied by difference gives first square minus second square.

The correct answer is B: 2pq. The side of the large square is split into lengths p and q in both directions. This creates two rectangular regions whose dimensions are p by q. Using the rectangle-area rule, length × breadth, the area of each rectangle is p×q=pq. Since there are two such rectangles, their combined area is pq+pq=2pq. Option A, p²+q², represents the two square regions, not the rectangles. Option C, pq, gives the area of only one rectangle. Option D, (p+q)², is the area of the entire large square, including all four regions. Memory cue: in (p+q)²=p²+2pq+q², the middle term 2pq comes from two equal p-by-q rectangles.

The correct answer is A. Begin with the large square, whose area is a². Removing one rectangular strip of area ab gives a²-ab. Removing the second strip gives a²-ab-ab=a²-2ab. However, the corner square of area b² lies in the overlap of the two removed strips, so it has been subtracted twice. It must therefore be added once, giving a²-2ab+b². This is exactly the expansion of (a-b)². Option B has plus signs and describes (a+b)², where regions are added. Option C is the difference-of-squares identity and has a different arrangement. Option D forgets the overlap correction. Memory cue: subtracting two overlapping regions requires adding their common part once.

The correct answer is D: (a+b)(a-b). The original large square has area a², and removing the small square of area b² leaves a²-b². The difference-of-squares identity states a²-b²=(a+b)(a-b). A visual cut-and-rearrange model shows that the remaining pieces can be rearranged into a rectangle whose two side lengths are a+b and a-b, without changing total area. Option A is the square of a+b and is generally too large; it represents a different identity. Option B includes only a and a+b, so it does not equal a²-b² in general. Option C has b as one factor and also gives the wrong area. Memory cue: difference of two squares always factors as sum × difference.

Direct answer: Option A, 10y. A square with side y+5 has area (y+5)². Apply the identity (a+b)²=a²+2ab+b² with a=y and b=5. The large square contributes y². The two equal cross-rectangles have areas y×5=5y and 5×y=5y, so together they contribute 5y+5y=10y. The small corner square has area 5²=25. Thus the full expansion is y²+10y+25, and the middle term is 10y. Option B is only the area of one cross-rectangle, so it misses the second identical part. Option C is the constant term produced by the 5-by-5 corner. Option D is the first term produced by the y-by-y square. The key visual idea is that squaring a binomial creates two cross-parts, which is why the coefficient is 2ab rather than ab. A quick memory cue is: in (a+b)², the middle term is twice the product of the two side parts.

The correct answer is C: r²+5r+6. The area of a rectangle is length × breadth, so calculate (r+2)(r+3). Multiply every term in the first bracket by every term in the second: r·r=r², r·3=3r, 2·r=2r, and 2·3=6. Adding these four regions gives r²+3r+2r+6=r²+5r+6. Option A omits both cross-products and keeps only r² and 6. Option B lists the two cross-products but omits the constant 6 and does not combine like terms. Option D has the right terms but incorrect coefficients: the cross-products total 5r, not 6r, and the constant is 6, not 5. Memory cue: for (r+m)(r+n), the expansion is r²+(m+n)r+mn.

The correct answer is A: 4x²+4x+1. A square’s area is the side multiplied by itself, so the area is (2x+1)². Apply (p+q)²=p²+2pq+q² with p=2x and q=1. Then p²=(2x)²=4x², the two cross-regions give 2pq=2(2x)(1)=4x, and q²=1. Adding these parts gives 4x²+4x+1. Option B incorrectly changes the square of 2x and the cross-term. Option C leaves out the two rectangular cross-regions, so it misses 4x. Option D uses 2x instead of 4x for the middle term. Memory cue: when squaring a binomial, never omit 2pq; it represents two equal rectangles in the area model.

The correct answer is B: a+b and a-b. The expression a²-b² is the difference between two squares. The difference-of-squares identity gives a²-b²=(a+b)(a-b). Therefore, when the remaining pieces are rearranged into one rectangle, its side lengths are a+b and a-b. Option A gives a rectangle of area ab, not a²-b². Option C treats the areas a² and b² as if they were side lengths; multiplying them would give a²b², not the required expression. Option D gives area 4ab, which is also different. The rearrangement changes the shape but preserves area, so the product of the new sides must equal the original difference. Memory cue: for two squares, factor as sum of roots multiplied by difference of roots.

The correct answer is C: k²-8k+16. The square’s area is (k-4)². Use (a-b)²=a²-2ab+b² with a=k and b=4. The large square contributes k². Two rectangular strips, each of area 4k, are removed, giving -4k-4k=-8k. The corner square of side 4 is counted back with area 4²=16. Thus the total is k²-8k+16. Option A has only -4k and forgets that there are two strips. Option B uses a plus sign, which belongs to (k+4)². Option D is the difference-of-squares expression k²-16, not the square of a difference; it omits the required middle term. Memory cue: (a-b)² has a negative middle term and a positive final square.

Direct answer: Option D is correct. Imagine a large square whose side is a+b. Its area is therefore (a+b)². Divide each side into lengths a and b. The large square is then split into one square of side a, with area a²; two rectangles with sides a and b, each having area ab; and one square of side b, with area b². Adding the four non-overlapping areas gives a²+ab+ab+b², which simplifies to a²+2ab+b². Thus (a+b)²=a²+2ab+b². Option A is the difference-of-squares identity and has no 2ab part. Option B wrongly omits the two rectangles. Option C has an incorrect middle term and sign. Memory cue: the square of a sum contains first square, twice the product, and second square.

Direct answer: Option B, 5x. Split the first factor into x and 4 and the second factor into x and 1. The rectangle model then has four smaller regions: x×x=x², x×1=x, 4×x=4x, and 4×1=4. The terms containing exactly x are x and 4x. Since x means 1x, these are like terms and can be combined: 1x+4x=5x. Therefore the complete product is x²+5x+4. Option A, 4x, includes only the 4-by-x rectangle and forgets the x-by-1 rectangle. Option C, x², is the quadratic term from the large x-by-x region, not the sum of the x-terms. Option D, 4, is the constant corner region and has no variable. The important rule is that like terms have the same variable raised to the same power; x and 4x are like terms, but x² and x are not. A memory cue is to inspect each rectangle before adding: variable terms come from the two cross-products.

Direct answer: Option A. The four regions have areas a², ab, ab, and b². Adding them gives a²+ab+ab+b²=a²+2ab+b². The outer shape is a square whose side is made from lengths a and b, so its side is a+b and its total area is (a+b)². Equating the total area with the sum of its parts gives (a+b)²=a²+2ab+b². B would require negative middle regions, which this ordinary area model does not show. C and D are equivalent difference-of-squares statements and would involve subtraction rather than two positive ab rectangles. The useful check is to count both ab regions before selecting the identity.

Direct answer: Option A, a²−b². Area is found by subtracting the removed part from the original figure. The larger square has side a, so its area is a·a=a². The removed square has side b, so its area is b·b=b². Therefore the remaining L-shaped area is a²−b². Option B, (a−b)², is the area of a different square whose side itself is a−b; it is not generally the same as the large square with a smaller square removed. Option C adds the two square areas instead of removing one. Option D, 2ab, is the middle part in a square-of-a-sum identity and does not describe this subtraction. Memory cue: remaining area = original area − removed area.

Direct answer: Option A, z+3. Compare the expression with (z+r)²=z²+2rz+r². The coefficient of z is 6, so 2r=6 and r=3. The constant term also confirms this because r²=3²=9. Hence z²+6z+9=(z+3)², and the complete square has side length z+3. In the tile model, the z² tile forms the large square, two z-by-3 tiles together contribute 6z, and the 3-by-3 tile contributes 9. Option B would give (z+6)²=z²+12z+36, so both the middle and constant terms are wrong. Option C would give z²+18z+81. Option D would give (z−3)²=z²−6z+9, with the opposite sign for the middle term. A reliable method is to halve the coefficient of z and then check its square: 6 divided by 2 is 3, and 3² equals 9. This two-step check prevents confusing z+3 with z+6 or z−3.

Direct answer: Option B, a²−25. The area of a rectangle is length multiplied by breadth, so the required area is (a+5)(a−5). This is the difference-of-squares identity (u+v)(u−v)=u²−v², with u=a and v=5. Therefore, (a+5)(a−5)=a²−5²=a²−25. The visual model also shows why the middle terms disappear: a×(−5) gives −5a, while 5×a gives +5a, and these equal opposite terms cancel. The remaining parts are a² and −25. Option A has the wrong sign and would not result from the product. Option C is (a−5)², which has a −10a middle term and represents a different product. Option D is (a+5)², which has a +10a middle term and is also different. The memory cue is: sum times difference gives difference of squares, so there is no middle term. Always multiply the actual length and breadth rather than adding them.

The governing idea is addition of areas and combination of like algebraic terms. In an area model, the two rectangles are separate parts, so their contributions are added rather than multiplied. Their areas are 7x and 7x, and both contain the same variable part x. Therefore, add their coefficients while retaining x: 7x+7x=(7+7)x=14x. Option C is correct. The answer is not 49x because that would incorrectly multiply 7 by 7 while leaving x unchanged; if multiplication were intended, the variable power would also need careful treatment. Option A gives only one rectangle’s area, not the total. Option D would represent a product of two x-length quantities, which is not what the stated area labels require.