Algebra tiles

B is correct. The rectangle's area is length × breadth, so it is (a + b)(a − b). Expanding gives a² − ab + ab − b²; the middle terms cancel, leaving a² − b². This is the difference-of-squares identity, assuming the dimensions are valid for the rectangle.

The correct answer is B: x + 4 and x + 5. In a tile model, the x² tile comes from x multiplied by x. The two groups of x-tiles must have coefficients whose sum is 9, and the unit tiles must have a product of 20. The pair 4 and 5 satisfies both tests: 4 + 5 = 9 and 4 × 5 = 20. Thus x² + 9x + 20 factors as (x + 4)(x + 5). Option A has constants 1 and 20, whose product is 20, but their sum is 21, so its middle term would be 21x. Option C has sum 12, so it would produce x² + 12x + 20. Option D gives a sum of 29 and product 180, so neither coefficient matches. A useful method is to list factor pairs of the constant term first, then select the pair whose sum equals the coefficient of x. This is why the tile model supports factorisation rather than merely matching one number.

The correct answer is A. This product has the form (u+v)(u−v), so use the difference-of-squares identity (u+v)(u−v)=u²−v². Here u=a and v=4, giving (a+4)(a−4)=a²−4²=a²−16. The tile model says the positive a² region remains while a square of side 4, and therefore area 16, is removed. Option B has the wrong sign because the 16-area tile is subtracted, not added. Option C is the expansion of (a−4)², which contains a −8a term and is a different product. Option D is the expansion of (a+4)², not the product of a sum and a difference. In the tile picture, the +4 and −4 rectangular parts cancel. Memory cue: same first term with opposite second signs means first square minus second square.

The correct answer is A. The expression a² − b² means the area of a square of side a after a square of side b has been removed. The remaining pieces can be cut or rearranged without changing their total area. They fit together to form a rectangle whose side lengths are a − b and a + b. Therefore, a² − b² = (a − b)(a + b). Option A describes the essential rearrangement step and preserves the area. Option B adds area instead of representing subtraction, so it changes the original expression. Option C is not the general rearrangement needed and may not produce the required factors. Option D adds the two square areas, giving a² + b² rather than their difference. Memory cue: difference of squares becomes a rectangle with “difference” and “sum” as its sides.

Correct answer: B, -36. In the signed area model, the constant corner is formed by multiplying the two constant side lengths. These constants are -4 and +9, so their product is (-4)(9)=-36. Therefore the corner consists of negative constant tiles with total value -36. The full expansion confirms this: (x-4)(x+9)=x^2+9x-4x-36=x^2+5x-36. Option A, 36, loses the negative sign. Options C and D come from adding or subtracting the constants, but a corner area is obtained by multiplication, not by addition. The x terms create the two linear regions, while the constant-by-constant product creates the constant corner. Memory cue: in (x+p)(x+q), the constant term is pq, including both signs.

The direct answer is A: the total area of the x-tiles is 2x. Multiply the two signed binomials by arranging their tile regions: (x+4)(x-2)=x·x+x·(-2)+4·x+4·(-2)=x^2-2x+4x-8. The two regions containing exactly one x factor are -2x and +4x. Combine their signed areas: -2x+4x=2x. Thus the x-tiles together represent 2x. B, 6x, adds the magnitudes 2x and 4x while ignoring the negative sign. C, -2x, counts only the x by -2 tile and leaves out the +4x tile. D, -8x, incorrectly treats the constant region 4·(-2)=-8 as an x-region. Signed tiles require the signs to be preserved. Memory cue: collect like tiles only after writing each sign and coefficient.

The correct answer is A: 9p. First recognize p² + 18p + 81 as (p + 9)², because p² + 2(p)(9) + 9² = p² + 18p + 81. In the tile model, the middle term 18p is split into two equal rectangular pieces. Therefore each rectangle has area 18p ÷ 2 = 9p. Option B is the total area of both middle rectangles, not the area of one. Option C incorrectly combines the constant 81 with p. Option D is a side length of the completed square, not an area. The visual meaning of 2ab is important: it represents two rectangles, each with area ab. Here a = p and b = 9, so each is 9p. Memory cue: divide the middle term by two when the model has two equal middle rectangles.

The direct answer is B: x+4. Imagine the central x by x square. Along each side, the arrangement adds two strips of width 2, one contribution on each relevant side, so the total increase in side length is 2+2=4. Thus the outside square has side x+4. Its area would be (x+4)²=x²+8x+16: the x² part is the central square, the four x by 2 strips together contribute 8x, and the four 2 by 2 corner squares together contribute 16. A would add only one width-2 contribution and therefore gives too small a side. C incorrectly doubles x and adds 2. D is an area expression, not a side length. Memory cue: add lengths along one outer edge; do not add areas when the question asks for a side.

The direct answer is B: x+6. The area of the rectangle equals the product of its side lengths. We need a factor that, when multiplied by x+5, gives x²+11x+30. Factor the constant 30: the pair 5 and 6 has product 30, and their sum is 11. Thus x²+11x+30=(x+5)(x+6). Expanding verifies it: x²+6x+5x+30=x²+11x+30. A would give (x+5)²=x²+10x+25, so both the middle and constant terms are wrong. C gives (x+5)(x+11)=x²+16x+55. D gives x²+35x+150. A reliable method is to use the constant term first: 5 times the missing number must be 30, so the missing number is 6; then confirm that 5+6=11.

The correct answer is A: p = 14. The constant term 48 is obtained from the product of the two constant side parts: 6 × 8 = 48. In the rectangle model, the two mixed rectangular regions have areas 6x and 8x. Adding them gives the middle term 6x + 8x = 14x. Comparing this with px shows that p = 14. Option B is the product 6 × 8, so it gives the constant term, not the coefficient of x. Option C is not the sum or product required by the model and does not produce the middle term. Option D doubles the correct sum; it would count the two mixed regions more than once. The factorized form is (x + 6)(x + 8) = x^2 + 14x + 48, which confirms the result. Memory cue: in (x + r)(x + s), the constant term is rs and the x-coefficient is r + s.

The direct answer is A: x²+10x+25. A trinomial is a perfect square when it has the form x²+2kx+k²=(x+k)². In option A, half of the middle coefficient 10 is 5, and 5²=25, exactly the constant term. Hence x²+10x+25=(x+5)². In an algebra-tile model, one x² tile, ten x-tiles, and twenty-five unit tiles can be arranged to make a square with side x+5. Option B fails because the constant would need to be 25, not 20. Option C fails because a middle coefficient of 5 would require a constant of (5/2)², not 25. Option D has no x-term, so it cannot form the indicated three-part square with side x plus a constant. Memory cue: halve the middle coefficient and square it; the result must equal the constant term.

The correct answer is C. A square with side x+y can be divided by the x and y boundaries in both directions. The x-by-x part is one x² tile. The x-by-y and y-by-x parts are two separate rectangles, each represented by an xy tile. The y-by-y part is one y² tile. Adding these regions gives x²+xy+xy+y²=x²+2xy+y², which is (x+y)². Option A has only one xy tile and misses one of the two cross-regions. Option B gives the y² tile a negative sign, although all lengths and areas in this square are positive. Option D has an extra x² tile. Memory cue: a binomial square has one first square, two cross-rectangles, and one second square.