Visual models of identities

The direct answer is C: x+8. Identify the perfect-square pattern x²+2xy+y². Here the first term is x², the last term 64=8², and the middle term 16x equals 2·x·8. Therefore x²+16x+64=x²+2·x·8+8²=(x+8)². In the visual model, the outer side of the complete square is x+8. A, x+16, would give x²+32x+256, so both the middle and constant terms are wrong. B, x+4, would give x²+8x+16. D, x−8, would give a negative middle term, x²−16x+64, opposite to the expression given. Thus C is the only matching side. Memory cue: take the square root of the constant term and check whether twice that number times x gives the middle term.

Direct answer: B, area parts. In a visual model for (a+b)², the large square is divided into four regions: one square of side a and area a², two rectangles with sides a and b and total area ab+ab=2ab, and one square of side b and area b². Thus every term represents an area, not merely a length, angle, or perimeter. A is wrong because a², ab, and b² are products of lengths and describe regions, although a and b themselves may be lengths. C is wrong because no angle is represented by these algebraic terms. D is wrong because perimeter measures boundary length, whereas these terms add to the total area. Memory cue: in an area model, multiply the two side lengths of each tile; the resulting products are area terms.

Direct answer: A, r+s. Divide each side of the large square into lengths r and s. The four regions are an r by r square of area r², two r by s rectangles whose combined area is 2rs, and an s by s square of area s². Therefore the whole area is r²+2rs+s², which equals (r+s)². The side of the complete square is consequently r+s. B would give (r−s)²=r²−2rs+s², so its middle term is negative and does not match the model. C is only a product of r and s; it is not the sum of the two side segments. D would produce a different square and different terms. Memory cue: a positive doubled middle term indicates the sum of the two component lengths.

Direct answer: B, 81+18+1. Apply (a+b)²=a²+2ab+b² with a=9 and b=1. The large square contains a 9 by 9 square with area 9²=81, two rectangles each having area 9×1=9, and a 1 by 1 square with area 1²=1. The two rectangles together contribute 9+9=18, so the total is 81+18+1=100. A counts only one 9 by 1 rectangle and misses the second one. B includes both middle rectangles. C leaves out both rectangular parts, so it is incomplete. D gives the side expression, not the area; a length must be squared to find a square’s area. Memory cue: the middle term in (a+b)² is doubled: 2ab, never just ab.

Direct answer: A, the rectangle (x+3)(x+4). In a rectangle model, split one side into x and 3 and the other side into x and 4. The small corner is formed by the two constant pieces, so its area is 3×4=12. The full expansion is x²+4x+3x+12=x²+7x+12, and the final constant term confirms the corner area. B is wrong because the two constant pieces are 12 and 12, giving 12×12=144. C gives 2×5=10, not 12. D gives 6×6=36. Notice that the corner area is found by multiplying the two pieces that meet there, not by adding them. Memory cue: in (x+p)(x+q), the corner is pq and the two side strips together give (p+q)x.

Direct answer: B, the square with side 2. The term x² represents a large square whose side is x. Subtracting 4 means removing a smaller square with area 4. Since the area of a square is side×side, its side must satisfy side²=4, so the positive side length is 2. This also appears in the identity x²−4=x²−2²=(x−2)(x+2). A is wrong because side 4 gives area 16, not 4. B is correct because 2²=4. C and D are factors in the resulting factorisation, not the side of the constant square that was removed. In a geometric interpretation, lengths are taken as non-negative. Memory cue: first rewrite the subtracted number as a square: 4=2².

Direct answer: C, 49 and 14x. Split each side of the square into lengths x and 7. The x by x part has area x², each x by 7 rectangle has area 7x, and the small 7 by 7 square has area 7²=49. There are two equal rectangles, so their combined area is 7x+7x=14x. A is wrong because it gives 7 instead of the small square’s area 49. B correctly squares 7 but counts only one rectangle. C includes both required results. D confuses the factors and incorrectly writes 49x for the rectangular total. The complete identity is (x+7)²=x²+14x+49. Memory cue: square the constant for the small square, then double its product with x for the two rectangles.

Direct answer: B, the difference-of-squares identity. The expression has the form a²−b², so use a²−b²=(a+b)(a−b). With a=12 and b=8, it becomes (12+8)(12−8)=20×4=80. This is faster than finding 144 and 64 separately and subtracting. A is wrong because a perfect-square identity is used for expressions such as (a+b)² or (a−b)², not a difference of two squares. B matches the exact signs and structure. C concerns cubes, such as a³±b³, so it does not apply. D is not a standard identity for this expression. Memory cue: two squares separated by a minus sign become their sum multiplied by their difference.

Direct answer: B, 7x. Make a rectangle with side partitions x and 1 in one direction and x and 6 in the other. The four areas are x·x=x², x·6=6x, 1·x=x, and 1·6=6. The two terms containing x are like terms, so combine them: 6x+x=7x. A is only the area of one rectangle and ignores the x by 1 part. B combines both linear regions correctly. C counts only the smaller linear region and omits 6x. D is not produced by the model; subtracting 6x and x would be inappropriate because both regions are positive and are being added. The full expansion is x²+7x+6. Memory cue: in (x+p)(x+q), the total x-term is (p+q)x.

Direct answer: A. Divide each side of the large square into lengths a and b. The resulting four regions are an a by a square with area a², an a by b rectangle with area ab, a second b by a rectangle with area ab, and a b by b square with area b². Adding them gives a²+ab+ab+b²=a²+2ab+b², which is exactly (a+b)². B has two a² squares instead of the two different cross-rectangles. C includes only one ab rectangle and therefore gives a²+ab+b², missing ab. D has the wrong number and type of square regions and omits both cross-rectangles. Memory cue: the 2 in 2ab comes from two identical rectangles, not from one rectangle being doubled by guesswork.

Direct answer: A, x²+8x. The area of the x by x square is x×x=x². The area of the x by 8 rectangle is x×8=8x. When the two non-overlapping pieces are joined, their areas add, giving x²+8x. B is wrong because it treats 8 as though it were a side of an 8 by 8 square; the rectangle has dimensions x and 8, so its area is 8x, not 64. C reverses the multiplication incorrectly and also makes the square contribution wrong. D adds the side lengths rather than calculating areas and therefore has the wrong type of quantity. A useful check is to assign x=2: the separate areas are 4 and 16, whose sum is 20; x²+8x also gives 20. Memory cue: square area is side squared, while rectangle area is length times width.

The correct answer is B, 10x. In an area model, x^2 is represented by a square of side x, and 100 is represented by a square of side 10 because 10^2=100. The middle term 20x must be made from two equal rectangular parts, so it is split as 10x+10x. Each rectangle therefore has one side x and the other side 10, giving area 10×x=10x. A is wrong because 20x is the combined area of both rectangles, not one rectangle. B is correct because two copies give 20x. C is wrong because 100x does not represent the middle term. D is wrong because two copies of 5x give only 10x. Memory cue: when the middle term is divided into two equal visual parts, halve its coefficient.

The correct answer is B. Since 25=5^2, x^2-25 means the area of a square of side x minus the area of a square of side 5. Begin with the large x-by-x square, whose area is x^2, and remove the smaller 5-by-5 square, whose area is 25. The remaining pieces can be rearranged into a rectangle with sides x+5 and x-5, proving x^2-25=(x+5)(x-5). A represents (x+5)^2, which is a sum-type square, not a difference. C represents 5x, only one product term. D represents (x-5)^2, which includes different terms. Memory cue: difference of squares is shown by subtracting one square from another, then rearranging the remainder.

The correct answer is C, four. Divide each side of the square of side a+b into segments a and b. The two divisions create four regions: an a-by-a square with area a^2, an a-by-b rectangle with area ab, a second b-by-a rectangle with area ab, and a b-by-b square with area b^2. Adding them gives a^2+ab+ab+b^2=a^2+2ab+b^2. A is wrong because there are two square regions but also two rectangular regions. B is wrong because it misses one of the equal rectangles. D is wrong because no fifth region is created by the two side divisions. Memory cue: a two-part side creates four area blocks: square, rectangle, rectangle, square.

The correct answer is A: 4x^2+12x+9. The two factors are identical, so their product is the square of one factor: (2x+3)(2x+3)=(2x+3)^2. Use the identity (a+b)^2=a^2+2ab+b^2. Here a=2x and b=3. First, a^2=(2x)^2=4x^2. Next, the two cross-products together give 2ab=2(2x)(3)=12x. Finally, b^2=3^2=9. Adding these parts gives 4x^2+12x+9. Option A is therefore correct. Option B incorrectly treats (2x)^2 as 2x^2; the coefficient 2 must also be squared. Option C has only 6x in the middle, which is ab rather than 2ab, so one cross-product has been missed. Option D contains the first and last terms but omits both cross-products, so it is incomplete. A useful check is to remember that a square of a binomial has three terms, not just the two squared terms. Memory cue: square the first term, add twice their product, then square the second term.

The correct answer is A. A square model with side a+b+c has three same-part-by-same-part regions: an a-by-a square of area a^2, a b-by-b square of area b^2, and a c-by-c square of area c^2. The remaining regions are rectangles formed by unlike parts: ab, bc, and ca. Each unlike pair appears twice, so together they contribute 2ab+2bc+2ca. B lists the types of rectangular products, not square parts. C lists their combined doubled areas, also not squares. D is only a length, not an area. Memory cue: same letters multiplied form square regions; different letters multiplied form rectangles.

The correct answer is A: 2x^2+5x. The area of a rectangle is length multiplied by breadth. Therefore, area=(2x+5)×x. Use the distributive property by multiplying x by every term inside the bracket: x(2x+5)=x(2x)+x(5)=2x^2+5x. The area model shows the same idea visually. Split the length 2x+5 into parts 2x and 5. The first smaller rectangle has dimensions 2x and x, so its area is 2x^2. The second has dimensions 5 and x, so its area is 5x. Adding the two smaller areas gives 2x^2+5x. Option B loses the factor x in the second part, so it should have 5x, not 5. Option C has the wrong coefficient for both terms and does not follow distribution. Option D adds expressions rather than multiplying length by breadth, so it does not represent area. Memory cue: in an area model, partition one side, multiply each part by the other side, and add the resulting areas.

The correct answer is B. A visual algebra model represents quantities using lengths, squares, and rectangles. A square of side x represents x^2, a rectangle with sides x and y represents xy, and separate regions can represent terms that are then added. This makes multiplication and identities meaningful rather than purely symbolic. A is wrong because the model is for understanding, not just memorising. C is wrong because its main purpose here is area, although boundaries may also be visible. D is wrong because multiplication is still being performed; the diagram simply displays it geometrically. B is correct because each algebraic product is connected to an area. Memory cue: in an area model, factors are side lengths and their product is the region’s area.

The correct answer is C: 120 square units. The expression is a difference of two squares, so use A^2-B^2=(A+B)(A-B). With A=11 and B=1, we obtain 11^2-1^2=(11+1)(11-1)=12×10=120. This identity explains the rectangle interpretation: the original difference of square areas can be rearranged into a rectangle whose side lengths are the sum and difference of the original side lengths. Direct calculation confirms the result: 11^2=121, 1^2=1, and 121-1=120. Option A, 100, would be 10×10 and ignores the factor 12. Option B, 110, is neither the subtraction result nor the product 12×10. Option D, 121, is only 11^2 and forgets to subtract 1^2. Thus C is supported both by the identity and by ordinary calculation. Memory cue: for a difference of squares, multiply the sum by the difference; do not subtract the bases before squaring.

The correct answer is A: x^2+10x+25. Begin by simplifying the side length: x+2+3=x+5. Since the figure is a square, its area is side×side, so the area is (x+5)(x+5)=(x+5)^2. Apply the identity (a+b)^2=a^2+2ab+b^2 with a=x and b=5. Then a^2=x^2, 2ab=2×x×5=10x, and b^2=25. Therefore the area is x^2+10x+25. In a visual model, this consists of one x-by-x square with area x^2, two rectangles of dimensions x and 5, each with area 5x, and one 5-by-5 square with area 25. Option B includes only one 5x rectangle and misses the second cross-part. Option C does not follow the binomial-square identity and combines the terms incorrectly. Option D omits the two cross-rectangles entirely. Memory cue: simplify the complete side first, then use square = side squared and remember the two equal cross-regions.

The correct answer is A. A fixed figure has one fixed area, regardless of how its parts are described. If one calculation writes the area as a sum of smaller regions and another writes it as a product of whole side lengths, both calculations describe the same quantity. Therefore the two algebraic expressions are equal. This is the geometric basis of an algebraic identity. B is wrong because areas are not always negative; ordinary lengths and regions have non-negative area. C is wrong because rearranging or renaming parts does not change the total area. D is meaningless: the figure remains the object being measured. A is correct because equality comes from comparing two descriptions of the same area. Memory cue: same figure, same area, different expression means an identity.

The correct answer is C: \(q^2-8q+16\). A square has area equal to the square of its side, so the required area is \((q-4)^2\). Use the identity \((a-b)^2=a^2-2ab+b^2\), with \(a=q\) and \(b=4\). The first term is \(q^2\). The middle term is \(-2\times q\times4=-8q\), because the two cross-products are both negative. The last term is \(4^2=16\). Hence \((q-4)^2=q^2-8q+16\). Option A misses the factor 2 in the middle term. Option B has the wrong positive sign and corresponds to \((q+4)^2\). Option D has an incorrect middle coefficient and an incorrect constant term. A useful check is to substitute \(q=4\): the original area is zero, and only option C gives \(16-32+16=0\). Remember that a subtraction inside a square gives a negative middle term, not simply \(-q\times4\).

The correct answer is A: \(s^2-9\). The area of a rectangle is the product of its length and breadth, so multiply \((s+3)(s-3)\). This is the difference-of-squares identity \((a+b)(a-b)=a^2-b^2\), with \(a=s\) and \(b=3\). Therefore, the area is \(s^2-3^2=s^2-9\). If expanded directly, \(s^2-3s+3s-9\) results; the two middle terms cancel, leaving \(s^2-9\). Option B has the wrong sign for the constant term. Options C and D are expansions of \((s-3)^2\) and \((s+3)^2\), respectively; those are squares of one binomial, not the product of conjugate binomials in the question. A helpful memory cue is that equal opposite additions and subtractions cancel the middle terms, and the remaining result is the first square minus the second square. Thus the rectangle area is uniquely option A.

The correct answer is A: \((a+b)^2=a^2+2ab+b^2\). The outside shape is a square whose side is made from two lengths, \(a+b\), so its total area is \((a+b)^2\). The internal pieces have areas \(a^2\), \(ab\), \(ab\), and \(b^2\). Adding them gives \(a^2+ab+ab+b^2=a^2+2ab+b^2\). The two equal rectangles are important because they create two copies of \(ab\), not one. Option B would require the two cross-regions to have negative area or to be subtracted, which does not match the stated model. Option C describes a product of conjugate binomials and would have one positive and one negative contribution, giving no two positive \(ab\) rectangles. Option D incorrectly omits the two cross-products. A reliable memory cue is: a plus-shaped division of a square contains two positive cross-rectangles, so the middle term must be \(+2ab\).

The correct answer is A. Divide each side of the large square, whose side is \(a+b\), into parts of lengths \(a\) and \(b\). The resulting grid has four regions: an \(a\)-by-\(a\) square with area \(a^2\), an \(a\)-by-\(b\) rectangle with area \(ab\), a second equal \(b\)-by-\(a\) rectangle with area \(ab\), and a \(b\)-by-\(b\) square with area \(b^2\). Thus the total area is \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Option B leaves out both cross-rectangles and therefore gives too little area. Option C has no basis in the side lengths and does not describe the four regions. Option D mentions the two rectangles but ignores the two square regions, so it is incomplete. The important visual habit is to count every region created by dividing both directions into \(a\) and \(b\).