Visual models of identities

Correct answer: A. Removing a square of side z from a square of side x+y

Direct answer: Option A is correct. The first three terms form a perfect square: x²+2xy+y²=(x+y)². Therefore the complete expression is (x+y)²−z². This is a difference of two squares, represented visually by starting with a square of side x+y and removing a smaller square of side z. Applying u²−v²=(u−v)(u+v), with u=x+y and v=z, gives (x+y−z)(x+y+z). Option B begins with (x−y)², which has a negative 2xy term. Option C contains too few regions to represent x²+2xy+y²−z². Option D makes a larger square and would introduce additional positive terms instead of subtracting z². The reliable sequence is: identify the perfect square, represent subtraction by removing a square, then use the difference-of-squares identity.

Correct answer: A. One p² square, one q² square, and two pq rectangles

Direct answer: Option A is correct. Divide each side of the large square, whose length is p+q, into lengths p and q. The four resulting regions are a p by p square with area p², a q by q square with area q², and two p by q rectangles, each with area pq. Their total is p²+pq+pq+q²=p²+2pq+q²=(p+q)². Option B has the wrong number of square regions. Option C incorrectly calls pq a square; its two side lengths need not be equal, and the remaining shapes do not have the stated dimensions. Option D assigns q² to rectangles that should have area pq. The central check is to count the two cross-rectangles separately because together they create the 2pq term.

Correct answer: A. (a−b)²=a²−2ab+b²

Direct answer: Option A is correct. Start with the a by a square, whose area is a². Removing a strip of width b along one side removes area ab, and removing an equal strip along the adjacent side removes another ab. The corner where the strips meet is b by b, so it was counted as removed twice. Adding it back once gives a²−ab−ab+b²=a²−2ab+b². The remaining figure has side length a−b, so its area is (a−b)². Option B describes adding strips rather than removing them. Option C is the difference of two complete squares and uses a different model. Option D forgets the overlap correction and is therefore missing the +b² term. Memory cue: for a square of a difference, subtract two cross-products and add back the corner square.

Correct answer: A. (2x+1) and (x+3)

Direct answer: Option A is correct. The area of a rectangle equals length times width, so multiply each pair and compare with 2x²+7x+3. For A, (2x+1)(x+3)=2x²+6x+x+3=2x²+7x+3, exactly the required area. Option B gives (2x+3)(x+1)=2x²+2x+3x+3=2x²+5x+3, so its middle coefficient is wrong. Option C gives x²+4x+3, whose leading coefficient is wrong. Option D gives (2x+7)(x+3)=2x²+6x+7x+21=2x²+13x+21, so both the middle and constant terms are wrong. The input key marked B, but direct expansion proves that B is not correct; the corrected answer is A. A useful checking method is to multiply every option rather than comparing only one term.

Correct answer: B. \((a-b)^2\)

Answer: B, \((a-b)^2\). Begin with the square of side \(a\), whose area is \(a^2\). Each removed strip has area \(ab\), so removing both seems to subtract \(2ab\). However, the two strips overlap in a square of side \(b\), whose area is \(b^2\). That overlap was subtracted twice, so it must be added once. Therefore the remaining area is \(a^2-2ab+b^2\). By the identity \((a-b)^2=a^2-2ab+b^2\), the model represents \((a-b)^2\). A is the difference of two squares, not this three-part removal. C has a positive middle term and represents addition. D omits the two cross terms. Memory cue: subtracting two equal strips creates a negative \(2ab\) term, while the overlap returns \(+b^2\).

Correct answer: B. \(16x\)

Answer: B, \(16x\). The larger square has area \((x+4)^2\), and the smaller square has area \((x-4)^2\). The required difference is therefore \((x+4)^2-(x-4)^2\). Use the identity \((p+q)^2-(p-q)^2=4pq\), with \(p=x\) and \(q=4\): the difference is \(4\cdot x\cdot4=16x\). Direct expansion confirms it: \(x^2+8x+16-(x^2-8x+16)=16x\). A is only half the result. C is the area of a square with side expression involving 4, not the requested difference. D is a constant and ignores \(x\). Memory cue: the difference between the squares of \(x+r\) and \(x-r\) is \(4xr\).

Correct answer: B. \(9-x\)

Answer: B, \(9-x\). First rearrange the terms into descending powers of \(x\): \(81-18x+x^2=x^2-18x+81\). Compare this with \((u-v)^2=u^2-2uv+v^2\). Here \(u=x\) and \(v=9\), because \(2\cdot x\cdot9=18x\) and \(9^2=81\). Hence the expression is \((x-9)^2\). A square has the same value when its side is negated, so \((x-9)^2=(9-x)^2\); the listed suitable side is \(9-x\). A would produce a positive middle term, \(x^2+18x+81\). C and D do not square to the given trinomial. Memory cue: for a negative middle term, use the difference pattern and take half of 18 for the constant side.

Correct answer: B. 28

Answer: B, 28. Write the first and last terms as squares: \(4x^2=(2x)^2\) and \(49=7^2\). A perfect square with a positive middle term must therefore be \((2x+7)^2\). Apply \((p+q)^2=p^2+2pq+q^2\): the middle term is \(2(2x)(7)=28x\). Thus the coefficient of \(x\) is \(k=28\). A, 14, omits the factor 2 in the cross term. C doubles the correct result and D incorrectly multiplies the constant terms. The positive sign matters: \((2x-7)^2\) would give \(-28x\), not a positive middle term. Memory cue: take the square roots of the end terms, then multiply them and double the result.

Correct answer: C. 12ab

Answer: C, \(12ab\). For a square with side \(2a+3b\), split each side into lengths \(2a\) and \(3b\). The two cross rectangles have dimensions \(2a\) by \(3b\). Each rectangle therefore has area \((2a)(3b)=6ab\). Since there are two identical rectangles, their total area is \(6ab+6ab=12ab\). Equivalently, the cross term in \((2a+3b)^2\\) is \(2(2a)(3b)=12ab\). A adds the coefficients without multiplying, and B gives the area of only one rectangle. D ignores both the coefficients and the fact that there are two rectangles. Memory cue: the cross area is always twice the product of the two split lengths.

Correct answer: A. Because (b^2) has been removed twice

Direct answer: A. Begin with the large a-by-a square, whose area is a^2. To create the smaller square of side a-b, one strip of width b is removed from the top and another strip of width b from the side. Each strip has area ab. However, the two strips overlap at one corner in a b-by-b square. That corner has area b^2 and is included in both removed strips, so it has been subtracted twice. The desired removal should count it only once. Adding b^2 back once corrects the double subtraction: a^2-ab-ab+b^2=a^2-2ab+b^2=(a-b)^2. Option A states this exact overlap reason. Option B is false because b^2 is generally positive. Option C is false because ab and b^2 are not generally equal. Option D reverses the construction; the original a^2 is the starting area, not an extra amount to subtract. Common warning: “add back” does not mean adding a new region; it corrects an overlap error.

Correct answer: B. Removing a square of side (b) from a square of side (a) and making a rectangle

Direct answer: B. A square of side a has area a^2, and a smaller square of side b has area b^2. Removing the smaller square leaves the region a^2-b^2. If the remaining pieces are cut and rearranged without changing their total area, they form a rectangle whose two side lengths are a+b and a-b. Its area is therefore (a+b)(a-b), proving the identity a^2-b^2=(a+b)(a-b). Option A concerns partitioning a different complete square and does not specifically create the difference-of-squares remainder. Option B describes both the subtraction and the rearrangement needed. Option C produces the addition pattern associated with a binomial square. Option D only joins line segments; line lengths alone do not establish an area identity. A reliable memory cue is: remove the small square first, then rearrange the leftover pieces into a rectangle; the rectangle’s sides are the sum and difference of the original side lengths.

Correct answer: C. They become (2xy) and (-2xy) and cancel

Direct answer: C. Multiply the two binomials term by term: x·x=x^2, x·(-2y)=-2xy, 2y·x=2xy, and 2y·(-2y)=-4y^2. In the rectangle model, the two middle rectangles therefore have signed areas -2xy and +2xy. Their magnitudes are equal but their signs are opposite, so their combined area is -2xy+2xy=0. Only x^2-4y^2 remains, which is the difference-of-squares identity. Option A incorrectly adds magnitudes and ignores the opposite signs. Option B has the wrong coefficient and power; the y-by-y corner is -4y^2, not 2y^2. Option C correctly identifies the two cancelling middle rectangles. Option D is impossible because x^2 is a separate corner square and is not cancelled. In signed area models, opposite-oriented or negative tiles cancel equal positive tiles. Memory cue: conjugates (u+v)(u-v) have equal and opposite middle terms.

Correct answer: B. 25x^2+20xy+4y^2

The correct answer is B. The area of a square is the square of its side, so the required expression is (5x+2y)^2. Use (p+q)^2=p^2+2pq+q^2 with p=5x and q=2y. The large square contributes (5x)^2=25x^2. The two equal cross-rectangles together contribute 2(5x)(2y)=20xy. The small corner square contributes (2y)^2=4y^2. Adding all regions gives 25x^2+20xy+4y^2. Option A includes only one cross-product, 10xy, and forgets the second rectangle. Option C does not square the terms correctly. Option D assigns the wrong coefficients to the mixed and y^2 terms. A useful memory cue is that the middle term in a square always has the factor 2: (p+q)^2=p^2+2pq+q^2.

Correct answer: A. (x-4)(x+3)

The correct answer is A. In a rectangle model, the four regions come from multiplying two binomials. The x^2 region comes from x times x. The -4x region comes from x times -4, while the 3x region comes from 3 times x. Finally, 3 times -4 gives -12. Therefore the side labels are (x-4) and (x+3), so the multiplication is (x-4)(x+3). Expanding confirms it: x^2+3x-4x-12=x^2-x-12. Option B would give positive 4x and positive 12, not the displayed signs. Option C would give -3x rather than +3x and positive 12. Option D does not produce the four given products. Always track both the numbers and their signs in signed area tiles.

Correct answer: A. A rectangle whose sides equal the sum and difference of the two square sides

The correct answer is A. Let the side of the larger square be a and the side of the smaller square be b, with a>b. Removing the b by b square from the a by a square leaves an L-shaped region of area a^2-b^2. By cutting one part of this L-shape and shifting it, the pieces can be rearranged into a rectangle. Its two side lengths are a+b and a-b, so its area is (a+b)(a-b), which equals a^2-b^2. Option B would have area (a+b)^2, which is too large and represents a different identity. Option C does not preserve the required rectangular area relationship. Option D describes a special kind of parallelogram, not the standard rearrangement. Memory cue: difference of squares becomes a rectangle with sum on one side and difference on the other.

Correct answer: B. 11x

The direct answer is B: 11x. In a rectangle model, split the first side into 2x and 1, and the second side into 3x and 4. This creates four smaller regions: (2x)(3x)=6x², (2x)(4)=8x, (1)(3x)=3x, and (1)(4)=4. The x-terms are the two regions containing exactly one factor x, namely 8x and 3x. Adding like terms gives 8x+3x=11x. Option A, 8x, counts only one cross-region and misses 3x. Option B is correct because it combines both linear regions. Option C, 6x, confuses 6x² with an x-term and also loses one x. Option D, 12x, does not come from any correct pair of products. Thus the complete expansion is 6x²+11x+4. Memory cue: the middle term in a binomial product comes from the two cross-multiplications.

Correct answer: B. 20x

The direct answer is B: 20x. A square with side 2x+5 is divided by separating each side into 2x and 5. The corner squares have areas (2x)(2x)=4x² and 5×5=25. The two remaining rectangles each have dimensions 2x and 5, so each has area (2x)(5)=10x. Since there are two such rectangles, their combined area is 10x+10x=20x. Option A gives the area of only one rectangle, not both. Option B correctly counts both middle rectangles. Option C, 25x, incorrectly combines the constant 5 with x rather than multiplying the actual side lengths. Option D doubles the correct total. The full identity is (2x+5)²=4x²+20x+25. Memory cue: in (a+b)², the two middle rectangles together produce 2ab.

Correct answer: A. Two subtracted rectangles, each with area 6x

The direct answer is A. To model (x−6)², begin with a square of side x and remove a strip of width 6 along one direction and another strip of width 6 along the other direction. Each removed rectangle has area x×6=6x, so together they contribute −6x−6x=−12x. The small corner where the strips overlap has area 6×6=36 and is added back, giving x²−12x+36. Option A correctly identifies both subtracted rectangles. Option B gives only the positive constant term 36, not the negative linear term. Option C represents the original x² square. Option D is incomplete because a side expression alone is not an area and does not show the two cross-regions. The identity is (x−6)²=x²−2(6)x+6². Memory cue: the negative middle term comes from two removed strips, while the corner square is positive because it is restored.

Correct answer: A. A square of side a+b divided into one a² square, one b² square, and two ab rectangles

The direct answer is A. Draw a large square whose side is a+b. Divide both its length and width into parts a and b. The four regions are an a² square, a b² square, and two rectangles, each with area ab. Therefore the total area is a²+ab+ab+b²=a²+2ab+b². This shows visually why simply adding a² and b² leaves out the two cross-regions. Option A includes every region of the large square and is correct. Option B concerns a difference of lengths and does not model the square of a sum. Option C changes the geometric figure and gives no explanation of the identity. Option D hides the internal parts, so it cannot reveal the missing middle term. Common confusion: (a+b)² is not a²+b²; squaring a sum creates two cross-products.

Correct answer: A. One a² square, one b² square, and two rectangles each with area ab

The direct answer is A. The large square has side a+b. Divide one side into lengths a and b, and do the same to the adjacent side. The partition produces four regions: the a by a corner is a², the b by b corner is b², and the two remaining a by b regions are rectangles with area ab each. Adding them gives a²+ab+ab+b²=a²+2ab+b². Option A lists exactly these four regions. Option B misses one ab rectangle, so it would give the wrong middle term. Option C invents four square regions; the cross-regions are rectangles, not generally squares. Option D has the wrong number of rectangles and therefore cannot represent the four-part division. Memory cue: a square of a sum has two corner squares and two equal cross-rectangles.

Correct answer: C. A square of area 81

The direct answer is C: a square of area 81. Split the 18x term into two equal rectangles, each with area 9x. Each rectangle can have side lengths x and 9. To complete the large square, the missing corner must then be a 9 by 9 square, whose area is 9²=81. Algebraically, x²+18x+81=(x+9)². Option A uses 18 directly and does not create two 9x rectangles. Option B squares 6, but half of 18 is 9, not 6. Option C correctly uses half the coefficient of x and squares it. Option D squares 18, which is much too large and corresponds to the common mistake of forgetting the half-step. Memory cue: for x²+bx, add (b/2)²; here (18/2)²=9²=81.

Correct answer: B. 9x²−4

The direct answer is B: 9x²−4. This product has the form (A+B)(A−B), where A=3x and B=2. In the visual model, the positive and negative cross-regions have equal area 6x but opposite signs, so they cancel. The remaining areas are (3x)(3x)=9x² and the signed corner 2×(−2)=−4. Therefore the total is 9x²−4. Option A incorrectly adds 4 instead of using the negative corner. Option B correctly applies the difference-of-squares identity A²−B². Option C uses 6x², although (3x)² is 9x². Option D retains a middle term even though the two cross-terms cancel. Memory cue: conjugate factors with the same first part and opposite second parts produce a difference of squares.

Correct answer: B. −10

The direct answer is B: −10. Split the first dimension into x and 5, and the second into x and −2. The constant corner is formed by multiplying the two constant pieces, 5 and −2. Its signed area is 5×(−2)=−10. Expanding confirms this: (x+5)(x−2)=x²−2x+5x−10=x²+3x−10, so the constant term is exactly −10. Option A, 10, ignores the negative sign in −2. Option B correctly preserves the signs of both constant pieces. Option C is the sum of the two constants, not their product. Option D is their difference and also does not represent the corner product. In signed area models, a negative piece is treated algebraically as a subtraction. Memory cue: the constant term always comes from constant × constant, with the signs included.

Correct answer: A. A square with side 2a−3b

The direct answer is A. Compare the expression with the identity (p−q)²=p²−2pq+q². Here p=2a and q=3b. Then p²=(2a)²=4a², q²=(3b)²=9b², and −2pq=−2(2a)(3b)=−12ab. Therefore 4a²−12ab+9b²=(2a−3b)², which is represented by a square whose side is 2a−3b. Option A matches all three terms. Option B would produce 16a²−72ab+81b², so its coefficients are wrong. Option C represents (2a+3b)(2a−3b)=4a²−9b² and has no middle term. Option D produces a²−6ab+9b². Memory cue: check the first and last terms as squares, then verify the middle term is minus twice their roots multiplied.

Correct answer: B. a²+ab+b²

The direct answer is B: a²+ab+b². The correct area model for a square of side a+b contains four parts: one a² square, two ab rectangles, and one b² square. Its correct total is a²+2ab+b². If the two equal ab rectangles are mistakenly counted only once, one ab contribution is omitted. The written area then becomes a²+ab+b². Option A is the correct complete area, but it is not the result of the stated mistake. Option B accurately represents the undercount. Option C is the difference-of-squares form and does not describe this square. Option D leaves out the b² corner as well, so it is an even larger error. Memory cue: the coefficient 2 in 2ab records the two congruent cross-rectangles; forgetting one changes 2ab to ab.