Algebra tiles

Direct answer: option B, \(x^2+7x+12\). Algebra tiles represent different terms by their shapes and sizes. One square tile labelled \(x^2\) contributes \(x^2\). Seven rectangular \(x\)-tiles contribute \(x+x+x+x+x+x+x=7x\). Twelve unit tiles contribute 12. Adding the contributions gives the total area \(x^2+7x+12\). Option A swaps the coefficients of the \(x\)-term and the constant term, so it does not match the stated numbers of tiles. Option C uses a negative \(7x\) term; that would require negative or subtraction tiles, not seven ordinary positive \(x\)-tiles. Option D treats seven \(x\)-tiles as seven \(x^2\)-tiles, confusing their shapes and areas. Memory cue: count each tile type separately, attach its label, then add: one \(x^2\), seven \(x\), twelve 1s.

The correct answer is A: (x+2)². One x² tile represents an x-by-x square. Four x tiles can be arranged as two rectangular strips, each with area 2x, and four unit tiles form a 2-by-2 corner square with area 4. These pieces fit together to make a larger square whose side is x+2. Algebraically, (x+2)²=x²+2(x)(2)+2²=x²+4x+4, exactly matching one x² tile, four x tiles, and four unit tiles. Option B would require eight x tiles and sixteen unit tiles because (x+4)²=x²+8x+16. Option C is not the correct expansion and does not show a square form. Option D would involve negative strips and is unsuitable for this positive tile arrangement. Memory cue: four unit tiles make a 2-by-2 corner, so the added side length is 2, not 4.

The correct answer is B: x+4. We look for a perfect-square expression. The constant 16 is 4², so the small corner tile has side 4. The expression x²+8x+16 matches the identity x²+2·x·4+4²=(x+4)². In the tile model, there is one x² tile, two x-by-4 rectangular tiles whose combined area is 8x, and one 4-by-4 tile with area 16. Therefore the complete square has side x+4. Option A would produce x²+16x+64, not the given expression. Option C gives incorrect cross and constant terms. Option D would produce a negative middle term, x²-8x+16. Memory cue: take the square root of the constant, then check that twice its product with x equals the middle coefficient.

Direct answer: Option A, two tiles. A square with side x+1 can be divided by marking lengths x and 1 along both directions. Its area is (x+1)². Multiplying the side lengths gives x²+2x+1. In the visual model, x² is one large square, 1 is one unit square, and the middle term 2x is formed by two rectangles: one has dimensions x by 1 across the top, and the other has dimensions 1 by x along the side. Therefore there are exactly two x-rectangle tiles. B counts only one direction and misses the second rectangle. C has no matching term in the expansion. D would produce a middle contribution of 4x, not 2x. Memory cue: in a square model, the two crosswise strips are equal, so the x contribution is doubled.

Correct answer: A, 7x. Signed tiles are added algebraically, so the combined area is 12x+(-5x). Because both terms are like terms, subtract their coefficients: 12-5=7, and keep the common factor x. Thus the result is 7x. The positive tile is larger in magnitude, so the final sign is positive. Option B, 17x, would result from adding 12 and 5 while ignoring the negative sign. Option C, -7x, reverses the sign even though the positive magnitude is greater. Option D multiplies the coefficients and also changes the expression into a quadratic term, which is not what addition of linear tiles does. Memory cue: opposite signed tiles cancel in pairs; after cancellation, the sign belongs to the tiles left over.

The correct answer is A: (x + y)² = x² + 2xy + y². Add the tile areas: one x² tile contributes x², two xy tiles contribute xy + xy = 2xy, and one y² tile contributes y². Together the pieces form a square whose side is x + y, so the total area is (x + y)². Option B has a negative middle term, which would require negative or subtractive xy tiles. Options C and D are equivalent statements of the difference-of-squares identity and involve subtraction, not two positive xy rectangles. The visual sign matters: two positive mixed tiles mean +2xy. A quick memory cue is “plus side, plus middle”; for a square of x + y, every displayed tile area is positive.

The correct answer is B: x+3. The x² tile is a square whose side is x, because x×x=x². The small square has area 9, so its side is 3, because 3×3=9. Each 3x rectangle can be formed with side lengths x and 3. When the x² square, the two 3x rectangles, and the 3×3 square are arranged together, they make a larger square with side x+3. Check the total area: x²+3x+3x+9=x²+6x+9=(x+3)². Option A uses 9 as a side instead of recognising that 9 is an area. Option C would produce (x−3)²=x²−6x+9, with a negative middle term. Option D does not produce the given tile areas. Memory cue: the constant corner area 9 has side √9=3, not 9.

The correct answer is B: x+2. The expression x²+4x+4 is a perfect-square trinomial. Since 4=2², the small constant square has side 2. The two mixed rectangles each have area 2x, so together they contribute 4x. Therefore the tiles fit together as an outer square whose side is x+2: (x+2)²=x²+4x+4. Option A would give (x+4)²=x²+8x+16, which is too large. Option C would give (x−2)²=x²−4x+4, so the middle sign would be wrong. Option D would give (2x+2)²=4x²+8x+4, so even the leading term is incorrect. A useful method is to take the square root of the constant term and then check whether twice that number times x gives the middle term. Here 2×2×x=4x.