Visual models of identities

The correct answer is C, (x + 5) and (x + 6). To find the rectangle’s sides, factor the expression by looking for two numbers whose product is 30 and whose sum is 11. The numbers 5 and 6 satisfy both conditions: 5 × 6 = 30 and 5 + 6 = 11. Therefore, x² + 11x + 30 = (x + 5)(x + 6). In a visual model, x² is the large square, 5x and 6x are the two rectangular strips, and 30 is the small corner rectangle. Option A gives 3 × 8 = 24 and 3 + 8 = 11, so its constant term is wrong. Option B gives 25, not 30. Option C satisfies both product and sum. Option D gives 30 as the product but 17 as the sum, so its middle term is wrong. Memory cue: for x² + bx + c, check both sum = b and product = c.

The correct answer is D, 7 units. Rewrite the expression as a perfect square: s² − 14s + 49 = s² − 2(s)(7) + 7² = (s − 7)². In the visual model, the constant term 49 represents the area of the small corner square. A square with area 49 has side length √49 = 7, taking the positive root because a length is positive. Option A, 14, is the coefficient of the middle term, not the side length. Option B, 49, is the small square’s area, not its side. Option C, 2, is the numerical factor in the identity and does not represent the cut-out side. Option D is correct because 7² gives the constant term 49 and 2 × s × 7 gives the middle term 14s. Memory cue: in (s − a)², the corner square has area a² and side a.

The correct answer is A, 4x² + 12x + 9. The area of a square is side², so the required expression is (2x + 3)². Use (a + b)² = a² + 2ab + b², taking a = 2x and b = 3. Then (2x)² = 4x², 2(2x)(3) = 12x, and 3² = 9. Adding the three parts gives 4x² + 12x + 9. Option A includes all three area pieces in the visual model: the large x-part square, the two cross strips, and the corner square. Option B incorrectly changes (2x)² to 2x². Option C uses 6x instead of the doubled cross-term 12x. Option D omits the cross-term altogether. Memory cue: square the first term, double the product, then square the second term.

The correct answer is B, 9a² − 12a + 4. Since both sides are equal, the shape is a square and its area is (3a − 2)². Apply (p − q)² = p² − 2pq + q² with p = 3a and q = 2. The first square is (3a)² = 9a²; the middle term is −2(3a)(2) = −12a; and the last square is 2² = 4. Thus the total is 9a² − 12a + 4. Option A omits the middle term, so it is incomplete. Option B has every term and the correct signs. Option C incorrectly squares 3a as 6a² instead of 9a². Option D has a positive middle term, which would belong to (3a + 2)². Memory cue: a minus inside a square creates a negative middle term, but the final square remains positive.

The correct answer is C, (2x − 3)(2x + 3). Recognize 4x² − 9 as a difference of two squares: 4x² = (2x)² and 9 = 3². Use the identity A² − B² = (A − B)(A + B). With A = 2x and B = 3, we obtain 4x² − 9 = (2x − 3)(2x + 3). Option A treats 4x as the square root of 4x², but (4x)² would be 16x², so it is incorrect. Option B uses 9 as a factor rather than its square root 3. Option C uses the correct roots of both square terms. Option D would expand to x² − 9, not 4x² − 9. Memory cue: first rewrite each term as a square, then use the minus factor and plus factor.

The correct answer is D, 16 square units. The side x + 4 can be viewed as an x-part plus a 4-part. When the square is split into visual pieces, the two edge strips have areas 4x and 4x, giving 8x altogether. The remaining small corner is a square whose side is 4, so its area is 4² = 16. This also agrees with the identity (x + 4)² = x² + 8x + 16. Option A, 8, is the coefficient of x in the total strip area, not the corner area. Option B, 4x, is the area of one strip. Option C, 16x, has an unnecessary x and does not represent a square with side 4. Option D is correct because the corner has dimensions 4 by 4. Memory cue: the constant term in (x + a)² is a², the corner-square area.

The correct answer is B, (x + 5)². Compare the expression with the identity (x + a)² = x² + 2ax + a². Here the constant term is 25, so a = 5 because 5² = 25. The middle term then becomes 2(x)(5) = 10x, exactly as required. Therefore x² + 10x + 25 = (x + 5)². In a visual model, x² is the large square, the two equal strips together contribute 10x, and the small corner square contributes 25. Option A would expand to x² − 10x + 25, with the wrong sign. Option C is a difference of squares and has no 10x term. Option D expands to x² + 20x + 100, so both its middle and constant terms are wrong. Memory cue: half the coefficient of x gives the number in the perfect square, and its square gives the constant term.

Direct answer: Option C, x² + 8x − 9, is correct. The area of a rectangle is length × breadth, so multiply (x+9)(x−1). Using the distributive property, x(x−1)+9(x−1)=x²−x+9x−9. Now combine like terms: −x+9x=8x, giving x²+8x−9. Option A is wrong because it treats the two middle terms as 10x; the signs must be respected. Option B has the wrong sign for the 8x term. Option D is not the result of all four products and incorrectly keeps 9x−1 together. Memory cue: multiply every term in the first bracket by every term in the second bracket, then combine only like terms.

Direct answer: Option B, 2ab+2bc+2ca, is correct. Divide each side of the large square into lengths a, b and c. The three square regions have areas a², b² and c². Each pair of different lengths creates two matching rectangles: two rectangles of area ab, two of area bc, and two of area ca. Adding every region gives (a+b+c)²=a²+b²+c²+2ab+2bc+2ca. Option A counts only one rectangle for each pair and misses its matching partner. Option C is the side length, not the sum of region areas. Option D is a product of three variables and does not represent any rectangular region in this two-dimensional model. Cue: every pair appears twice because the area model has a rectangle above and below the corresponding diagonal arrangement.

Direct answer: Option A, one more pq rectangle, is correct. A square whose side is p+q can be divided into four regions: a p² square, a q² square, and two rectangles, each with area pq. Therefore (p+q)²=p²+pq+pq+q²=p²+2pq+q². The model already contains p², q² and one pq region, so exactly one further pq region is needed. Option B is wrong because p² is already present. Option C is wrong because q² is also already present. Option D describes a length, not an area region, and cannot supply the missing term. Memory cue: the middle term 2pq means two separate pq rectangles in the area model.

Direct answer: Option B, 6x, is correct. Begin with a square of side x, whose area is x². To change each side from x to x−6, a strip of width 6 and length x is removed along each direction. The area of one such strip is length × width=x×6=6x. The two strips together contribute 12x before the corner square of area 36 is considered, but the question asks for each strip separately. Option A gives only the width and is not an area. Option C is the combined area of two strips, not one strip. Option D is the 6×6 corner square, not a strip. A useful warning is that the two strips are each 6x, although their combined contribution is 12x.

Direct answer: Option A, (x+5) and (x−3), is correct. Multiply the proposed sides: (x+5)(x−3)=x²−3x+5x−15=x²+2x−15. Another quick check is to find two numbers whose product is −15 and whose sum is 2. Those numbers are 5 and −3. Option B has the correct product −15, but its numbers add to −2, giving x²−2x−15. Option C has product −15 but sum 14, so it gives x²+14x−15. Option D has product −30 and sum −13, so neither coefficient matches. In factorisation, always check both conditions: the constants multiply to the final constant and add to the coefficient of x.

The governing concept is the visual form of the identity (p + q)² = p² + 2pq + q². A square whose side is divided into lengths p and q contains one p-by-p square, two p-by-q rectangles, and one q-by-q square. Here p = 2a and q = 5. The first square has area (2a)² = 4a². Each rectangle has area (2a)(5) = 10a, so the two rectangles together have area 20a. The final small square has area 5² = 25. Therefore the model shows 4a², 20a, and 25, so option B is correct. Option A counts only one rectangle; C gives an incorrect first area; and D gives an incorrect constant and middle contribution.

The direct answer is B, −10x. The square has area (5x−1)². Apply (a−b)²=a²−2ab+b² with a=5x and b=1. The first square is (5x)²=25x². The two cross-products are −5x and −5x, so together they give −10x. The small square contributes 1²=1. Thus the complete expansion is 25x²−10x+1, and its middle term is −10x. Option A, 10x, would be the middle term of (5x+1)², where the sign is positive. Option C counts only one cross-product and forgets that the visual square contains two equal cross-regions. Option D, 25x, is incorrect in both coefficient and form; the first term is 25x², not 25x. The minus sign is essential because the binomial has subtraction. Memory cue: for (a−b)², the middle term is always −2ab.

Direct answer: Option A is correct. Since each side is split into lengths a and b, the whole square has side a+b and area (a+b)². Its four regions are an a² square, a b² square, and two rectangles of area ab. Adding them gives a²+ab+ab+b²=a²+2ab+b². Option B describes a difference model, where the cross-regions would have negative signs or be removed. Options C and D are equivalent difference-of-squares identities, involving subtraction and a rectangle with sides a+b and a−b; they do not match the four positive regions described here. Memory cue: two positive ab rectangles mean the square-of-a-sum identity and the middle term +2ab.

The direct answer is B. Start with a large square of area a^2. To represent a square whose side is a-b, remove a strip of area ab from one side and another strip of area ab from the adjacent side. This gives a^2-ab-ab=a^2-2ab. However, the small corner where the two strips meet has area b^2. That corner lies in both removed strips, so it has been subtracted twice even though it should be removed only once. We therefore add it back once, obtaining a^2-2ab+b^2. A is wrong because the corner is not simply left untouched; it is counted twice during subtraction. B correctly describes the correction. C is false because b^2 is zero only when b=0, not always. D is wrong because the expression uses addition and subtraction of areas, not multiplication of a^2 by b^2. Memory cue: in a difference-square model, the corner correction produces the final positive b^2 term.

The direct answer is A. A square with side x+7 has area (x+7)^2. Expanding it gives x^2+2(x)(7)+7^2=x^2+14x+49. In an area model, x^2 is the large square, 49 is the smaller square, and the middle term 14x is made from two congruent rectangles. Each rectangle has dimensions x and 7, so each has area 7x. Adding them gives 7x+7x=14x. A therefore describes the two equal strips correctly. B would give 28x, not 14x, so it is too large. C gives 98x and uses the wrong dimensions. D names the two square regions, not the two equal rectangular strips. A useful memory cue is that the middle term in (x+k)^2 is 2kx: it represents two equal kx rectangles in the tile model.

The direct answer is C. To find the side of a square from its area expression, look for a perfect-square identity. The first term x^2 is x squared, so x is one part of the side. The last term 9 is 3^2, so the other part has magnitude 3. The middle term in (x-3)^2 is -2(x)(3)=-6x, which matches the given expression. Therefore x^2-6x+9=(x-3)^2, and the side is x-3. A gives (x+3)^2=x^2+6x+9, with the wrong sign in the middle. B would produce (x-6)^2=x^2-12x+36, so both the middle and constant terms are wrong. D would produce x^2+18x+81, which does not match. The sign of the middle term is important: a negative middle term means the binomial contains subtraction. Memory cue: identify the square roots of the first and last terms, then check twice their product.

The direct answer is D. The area of a rectangle is length multiplied by width, so calculate (2x+1)(2x-1). These are conjugate binomials: they have the same first term and opposite signs in the second term. The identity (a+b)(a-b)=a^2-b^2 applies with a=2x and b=1. Thus the area is (2x)^2-1^2=4x^2-1. If expanded directly, the products are 4x^2-2x+2x-1; the middle terms cancel, again giving 4x^2-1. A incorrectly adds the squares instead of taking their difference. B is the expansion of (2x+1)^2, not the product of the two different sides. C is the expansion of (2x-1)^2, so it also squares only one binomial. D correctly represents the rectangle. Memory cue: same terms with opposite signs give a difference of squares, and the cross terms cancel.

The direct answer is A. Imagine a large square whose side is p+q. Divide each side into lengths p and q. The large square is then divided into four parts: one p by p square with area p^2, one q by q square with area q^2, and two rectangles. Each rectangle has one side p and the other side q, so each area is pq. Their combined area is pq+pq=2pq, giving the identity (p+q)^2=p^2+2pq+q^2. Therefore the two pq rectangles are always equal in area because they have the same dimensions, even if p and q are different. B is not always true: p^2 and q^2 are equal only when the magnitudes of p and q are equal. C and D compare a square with a rectangle, whose areas are not generally equal. Memory cue: the middle term 2pq comes from two matching cross-rectangles.

The direct answer is B. The square has side a+2b, so its area is (a+2b)^2. Use the identity (x+y)^2=x^2+2xy+y^2 with x=a and y=2b. The middle term is 2xy=2(a)(2b)=4ab. In a visual model, this term comes from two equal rectangles: each has dimensions a and 2b and therefore area 2ab; together they give 2ab+2ab=4ab. A forgets the second rectangle or fails to include the coefficient 2 in 2b. C is only twice b^2 and is not the cross term. D contains the two square areas a^2 and 4b^2 but omits the two rectangles, so it is incomplete. Memory cue: in a binomial square, the middle term is twice the product of the complete two terms, not just twice ab.

The direct answer is C. Recognize the first and last terms as squares: 16p^2=(4p)^2 and 9=3^2. The negative middle term shows that the two parts are subtracted. Applying (u-v)^2=u^2-2uv+v^2 with u=4p and v=3 gives (4p-3)^2=(4p)^2-2(4p)(3)+3^2=16p^2-24p+9. A uses addition, so its middle term would be +24p. B squares 16p rather than 4p and would produce 256p^2-96p+9. D has first term p^2, not 16p^2, and therefore cannot match. The sign and square roots of the first and last terms are enough to identify the correct binomial, but checking the middle term confirms it. Memory cue: a perfect square with a negative middle term has the form (first square root minus last square root)^2.

The direct answer is D. The rectangle’s area is the product of its side lengths: (3x+2)(3x-2). The factors have the form (a+b)(a-b), where a=3x and b=2. By the difference-of-squares identity, (a+b)(a-b)=a^2-b^2. Therefore the area is (3x)^2-2^2=9x^2-4. Direct multiplication gives 9x^2-6x+6x-4; the two middle terms cancel, giving the same answer. A incorrectly adds the squares. B is (3x-2)^2, and C is (3x+2)^2; both square only one side instead of multiplying the two different sides. D correctly uses the conjugate pair. Memory cue: equal first parts and opposite second signs mean “square the first, subtract the square of the second.”

The direct answer is A. Take a large square with side a+b and divide each side into lengths a and b. This creates four regions: an a by a square with area a^2, a b by b square with area b^2, and two congruent rectangles, each measuring a by b and therefore having area ab. Adding all four non-overlapping parts gives a^2+ab+ab+b^2=a^2+2ab+b^2. This is the identity (a+b)^2. B describes a difference of squares, which involves removing regions rather than adding all four parts. C counts only one of the two ab rectangles, so it misses part of the model. D has a negative middle term and corresponds to (a-b)^2, not a large square split using a+b. Memory cue: two matching cross-rectangles always contribute 2ab in a complete square area model.

The direct answer is B: -12. In an algebraic product, the constant term is obtained by multiplying the terms that contain no variable. The two factors are (x-2) and (x+6). Their constant parts are -2 and +6, so their product is (-2)(6)=-12. To check completely, distribute each term: x·x=x², x·6=6x, (-2)·x=-2x, and (-2)·6=-12. Combining the middle terms gives x²+4x-12, so the constant term is -12. Option A, 12, incorrectly ignores the negative sign in -2. Option C, 4, is the coefficient of x after combining 6x and -2x, not the constant term. Option D, -4, does not result from multiplying the constant parts. Memory cue: constant times constant gives the constant term; always carry the signs.