Visual models of identities

The correct answer is A: (x+6)(x-5). To factor x^2+x-30, find two numbers whose product is -30 and whose sum is +1, because the middle coefficient is +1. The numbers are +6 and -5: 6 times -5=-30 and 6+(-5)=1. Therefore x^2+x-30=(x+6)(x-5). In a rectangle model, the x^2 tile supplies the variable parts, the signed corner tiles multiply to -30, and the two middle strips combine to +x. Option A satisfies both checks. Option B also has product -30, but its constants add to -1, giving x^2-x-30, so its middle sign is wrong. Option C has constants with product -30 but sum 7, giving x^2+7x-30. Option D has product -30 but sum -29, not +1. Memory cue: for a monic quadratic, check both the product of constants and their sum before accepting the factorisation.

The correct answer is A: a^2+b^2+2ab. The complete expression is (a+b+c)^2. Its pieces are three corner squares a^2, b^2, c^2 and three pairs of strips: 2ab, 2ac and 2bc. If every part involving c is removed, remove c^2, 2ac and 2bc. The remaining pieces are a^2+b^2+2ab. These pieces form the square identity (a+b)^2, because a^2+2ab+b^2=(a+b)^2. Option A includes exactly all parts made from a and b. Option B incorrectly keeps c^2 even though the c corner is removed and also omits the ab strips. Option C lists the removed c-related area rather than the area left behind. Option D includes only one of the two ab rectangles, so it is incomplete. Memory cue: after removing all c-parts, simply set c=0 in (a+b+c)^2; the result is (a+b)^2.

The direct answer is B. To build a rectangle from 2x^2+7x+3, split the middle term into two terms so that their sum is 7x and the factor grouping works with 2x^2 and 3. The numbers 1 and 6 add to 7, and their product is 6; this agrees with the product of the outside coefficients 2×3=6. Therefore 2x^2+7x+3=2x^2+x+6x+3. Group the visual tiles: (2x^2+x)+(6x+3)=x(2x+1)+3(2x+1)=(2x+1)(x+3). A uses 2 and 5, whose product is 10, not 6. C uses 3 and 4, whose product is 12, so it does not make equal factors. D breaks the middle term into a constant 7 and does not preserve like terms. Thus B gives the correct rectangle construction. Memory cue: for ax^2+bx+c, find two middle coefficients with sum b and product ac.

The direct answer is B: the two negative strips together have total area -12x. Expand the product carefully: (x-5)(x-7)=x·x+x(-7)+(-5)x+(-5)(-7)=x^2-7x-5x+35=x^2-12x+35. In a signed tile model, the x by x tile is positive, each strip involving one negative constant is negative, and the corner formed by two negative constants is positive. Thus the strips have signed areas -7x and -5x, and they add to -12x. Option A has the wrong sign because the strip lengths are negative. Option B correctly represents both strips and their sum. Option C confuses the constant corner 35 with a strip and incorrectly writes 35x. Option D would require one positive and one negative strip of equal or opposite contribution; here both strips are negative, so they do not cancel. Memory cue: one negative factor makes a negative strip; two negative factors make a positive corner.

The direct answer is C: the sides are x-5 and x-7. For a quadratic x^2+px+q, look for two numbers whose sum is p and whose product is q. Here the required numbers must add to -12 and multiply to 35. The pair -5 and -7 works because -5+(-7)=-12 and (-5)(-7)=35. Therefore (x-5)(x-7)=x^2-7x-5x+35=x^2-12x+35. In the tile model, x^2 is the large tile, the two negative strips give -12x, and the positive corner gives 35. Option A gives x^2-36x+35, so its middle term is wrong. Option B gives x^2+12x+35, with the wrong sign in the middle. Option C matches both the middle and constant terms. Option D gives x^2-23x-420, so both important coefficients are wrong. A useful cue is that a positive constant with a negative middle coefficient usually requires two negative factor numbers.

The direct answer is D: the common error is failing to add b^2 back. Begin with the large a by a square, whose area is a^2. Removing one a by b strip subtracts ab, and removing the second strip subtracts another ab, giving a^2-2ab. However, the two strips overlap at the small b by b corner. That corner has been removed twice, although it should be absent only once. To correct this double subtraction, add b^2 back. The final result is a^2-2ab+b^2=(a-b)^2. Option A uses the plus-side model and would lead to a positive middle term. Option B counts the large square twice, which is not what the model shows. Option C misses the fact that the strips are being removed, so their contribution must be negative. Option D correctly identifies the over-subtraction of the corner. Memory cue: when two removed regions overlap, add their overlap once to repair double subtraction.

The direct answer is B: the second area is greater by (a-b)^2/4. Expand the first expression: (x+a)(x+b)=x^2+(a+b)x+ab. For the second expression, use the square identity: (x+(a+b)/2)^2 = x^2+(a+b)x+(a+b)^2/4. Both expressions contain exactly the same x^2 term and the same linear term (a+b)x, so those parts cancel when we compare areas. The difference is (a+b)^2/4-ab. Put over a common denominator: [(a+b)^2-4ab]/4 = [a^2+2ab+b^2-4ab]/4 = (a^2-2ab+b^2)/4 = (a-b)^2/4. Option A is only a length-like quantity, not the area difference. Option B gives the complete remaining constant area. Option C subtracts the wrong constant terms and ignores the square model. Option D does not simplify to the required difference. The result is always nonnegative because it is a square divided by 4.

The correct answer is B: a² − 2ab + b². Begin with the full square area a². Removing the first strip subtracts ab, and removing the second strip subtracts another ab, giving a² − 2ab. However, the corner of area b² belongs to both removed strips, so it has been subtracted twice. To correct double subtraction, add b² back once. The result is a² − 2ab + b², which is also (a − b)². Option A subtracts the overlap one time too many. Option C adds the strip areas instead of removing them. Option D removes only the difference of two squares and does not represent this two-strip counting process. The key rule is: when an overlapping region is removed twice, add it back once.

The correct answer is C: −12ab. The square area is (3a − 2b)². Use (u + v)² = u² + 2uv + v² with u = 3a and v = −2b. The middle term is 2uv = 2(3a)(−2b) = −12ab. The complete expansion is 9a² − 12ab + 4b². In the visual model, each mixed rectangle contributes (3a)(−2b) = −6ab, and the two equal rectangles together contribute −12ab. Option A has the wrong sign. Option B is only the magnitude of one mixed contribution doubled incorrectly. Option D is the area of one mixed rectangle, not both. The negative sign comes from subtracting 2b in the side.

The correct answer is A: a = 11 and b = 4. The difference-of-squares identity gives a² − b² = (a + b)(a − b). Therefore the two rectangle sides are a + b and a − b. Using the given sides, a + b = 15 and a − b = 7. Add the equations: 2a = 22, so a = 11. Subtract them: 2b = 8, so b = 4. Check: 11 + 4 = 15 and 11 − 4 = 7. Option B uses the rectangle sides directly as a and b, but then a + b and a − b would not be 15 and 7. Option C gives a + b = 15 but a − b = 1. Option D gives a + b = 30 and a − b = 14. The safe method is to use half the sum and half the difference.

The direct answer is C: 2xy. The complete square has area (x+y)^2. Its parts are two small squares with total area x^2+y^2 and two remaining rectangles. Therefore, the rectangles together have area (x+y)^2-(x^2+y^2). Expand the square: x^2+2xy+y^2. After subtracting x^2+y^2, only 2xy remains. Equivalently, each rectangle has side lengths x and y, so each has area xy; two such rectangles give xy+xy=2xy. A is only the area of one rectangle, not both. B is the whole outer square, so it includes the two small squares as well. D is the difference of two squares and does not represent the two cross-rectangles in this model. Memory cue: in (x+y)^2, the two cross-parts together always produce 2xy.

The direct answer is B. Imagine a square whose side is divided into lengths a, b, and c. The partition contains three square regions, a^2, b^2, and c^2. For every pair of different lengths, there are two rectangles: two ab rectangles, two bc rectangles, and two ca rectangles. Their combined area is ab+ab+bc+bc+ca+ca=2ab+2bc+2ca. Thus the full identity is (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca. A counts only the three square regions and misses all cross-rectangles. C refers to a cube, but this is a two-dimensional area model. D concerns boundary length, not area. Memory cue: every unordered pair among a, b, c appears twice in a square partition.

The direct answer is A: the constant corner is 7 and the combined x-term is 16x. Split the rectangle into four products: (2x)(2x)=4x^2, (2x)(7)=14x, (1)(2x)=2x, and (1)(7)=7. The two cross-products are like terms, so 14x+2x=16x. The constant corner comes from multiplying the two constant pieces, 1 and 7, giving 7. Therefore the full area is 4x^2+16x+7. B incorrectly treats the constant as 14 and gives the wrong linear total. C has the correct constant but incorrectly says 8x. D uses 1 instead of 1×7 and omits the 2x cross-product. Memory cue: in a rectangle model, multiply every length part with every part on the other side, then combine only like terms.

The direct answer is C: 5x-3y. Use the identity (p-q)^2=p^2-2pq+q^2. Taking p=5x and q=3y gives (5x-3y)^2=(5x)^2-2(5x)(3y)+(3y)^2=25x^2-30xy+9y^2, exactly the given area. The negative middle term tells us that the two parts of the side are being subtracted. A gives (5x+3y)^2, whose middle term would be positive 30xy. B is not the square root of the expression and does not produce the given three terms. D is twice the correct side, so its square would be four times the area. Memory cue: positive middle term suggests a plus binomial square; negative middle term suggests a minus binomial square.

The direct answer is C: 12. The expression a^2+2ab+b^2 is the perfect-square identity (a+b)^2. Therefore the model says (a+b)^2=144. Taking square roots gives a+b=12 or a+b=-12, but the question states that a+b is positive, so we choose 12. In a square, area equals side squared, so the positive side length is the square root of 144. A is not the square root of 144; 24^2 is 576. B is half of 144, not the required side. D is the area itself, not its side. The positivity condition removes any ambiguity about the sign. Memory cue: for a square model, side is the positive square root of area; first identify the perfect-square expression.

The direct answer is B: 65. The difference-of-squares identity is a^2-b^2=(a+b)(a-b). The rectangle model has one side a+b and the other side a-b, so its area is the product of these two given lengths. Substitute the values: (13)(5)=65. There is no need to solve separately for a and b. A, 18, is the sum 13+5 and is not the product required by the rectangle area. C, 169, is 13^2 and ignores the second factor. D, 25, is 5^2 and ignores the first factor. The condition that the model is a rectangle reinforces multiplication of the two side expressions. Memory cue: conjugate factors (a+b) and (a-b) multiply to a^2-b^2 because the two middle products cancel.

The direct answer is A. Add the four regions: the first square contributes a^2, the second contributes b^2, and the two congruent rectangles contribute ab+ab=2ab. Thus the whole square has area a^2+2ab+b^2. Its side is made by joining lengths a and b, so the same whole area is (a+b)^2. Hence (a+b)^2=a^2+2ab+b^2. B would require negative rectangular regions or a subtraction arrangement, not two positive rectangles. C describes a rectangle with sides a+b and a-b, producing a^2-b^2. D incorrectly leaves out the 2ab contribution and is generally false. Memory cue: two positive ab rectangles mean a positive 2ab middle term.

The direct answer is A. Begin with an x by x square of area x^2. To shorten both dimensions by 4, remove one strip of dimensions x by 4, area 4x, and another strip of dimensions 4 by x, also area 4x. The corner where the strips overlap is a 4 by 4 square of area 16. It was removed twice, so it must be added back once. The resulting area is x^2-4x-4x+16=x^2-8x+16=(x-4)^2. B uses incorrect areas and does not represent the two strips. C reverses the removal and addition. D removes the original square itself, which destroys the model. Memory cue: in a difference square, subtract two strips and add back their overlap once.

The direct answer is C: x^2-y^2. The rectangle area is length times breadth, so it is (x+y)(x-y). Multiply each term: x·x=x^2, x·(-y)=-xy, y·x=xy, and y·(-y)=-y^2. The middle terms -xy and +xy cancel, leaving x^2-y^2. This is the difference-of-squares identity. A would be obtained only if both dimensions were x-y. B omits the effect of the cross-products and the negative y^2 term. D incorrectly keeps an extra 2xy and also has the wrong final sign. The condition x>y ensures x-y is positive, so the dimensions can represent a usual rectangle. Memory cue: conjugate binomials have opposite signs, so their middle terms cancel.