Visual models of identities

The correct answer is B: 10x. In a square model for (x + 10)^2, the four parts are an x^2 square, two equal rectangles, and a 100 square. Indeed, x^2 + 20x + 100 = x^2 + 2(x)(10) + 10^2. The total middle term 20x is made from two equal rectangles, so each rectangle has area 20x ÷ 2 = 10x. Option A is the combined area of both rectangles, not the area of each one. Option B is correct because 10x + 10x = 20x. Option C uses the corner area 100 as if it were a rectangle area; its side length is 10, not 100. Option D is too small and would produce only 10x in total. The key distinction is between the total middle term and one of its two equal parts. Memory cue: in (a+b)^2, each middle rectangle has area ab, while both together make 2ab.

The governing idea is the rectangle or factor model for a quadratic expression. In x²+13x+40, the two smaller side lengths must multiply to the constant term 40, while their sum must produce the coefficient 13 of x. The pair 5 and 8 satisfies both relationships: 5×8=40 and 5+8=13. In the visual model, the product 5×8 represents the area of the small corner rectangle, and the two side contributions together create the 13x part. Option B is incorrect because 4×10 equals 40, not 13. Options C and D do not form the required factor pair. Therefore, option A is correct.

The correct answer is A: the square with side a. Divide the large square of side a+b into four parts by drawing one line after length a and another after width a. One part is a square with length a and breadth a, so its area is a × a = a^2. Another part is a square of side b with area b^2. The remaining two parts are rectangles with dimensions a and b, each having area ab; together they give 2ab. Option A identifies the source of a^2 exactly. Option B gives b^2, not a^2. Option C gives the middle term 2ab, not the first square term. Option D refers to boundary length, whereas a^2 is an area and cannot come directly from the perimeter. This partition is the visual proof of (a+b)^2 = a^2 + 2ab + b^2. Memory cue: each squared term comes from a small square; the mixed term comes from the two rectangles.

The correct answer is B: 4xy. The area of the square is (x + 2y)^2. Use the identity (a+b)^2 = a^2 + 2ab + b^2, with a = x and b = 2y. The combined middle term is 2ab = 2 × x × 2y = 4xy. In the visual model, there are two equal rectangles; each has dimensions x and 2y, so each area is 2xy. Adding the two gives 4xy. Option A is the area of only one rectangle, because it omits the second rectangle. Option B correctly includes both. Option C is part of the final square term only if the expression is handled differently; it is not the mixed term. Option D is 4y^2, which has no factor x and represents neither the middle rectangles nor the correct square term. Be careful to use the complete term 2y as b, not just y. Memory cue: the middle term is twice the product of the complete two side pieces.

The correct answer is A: x^2 - 14x + 49. Use the square-of-a-difference identity (a-b)^2 = a^2 - 2ab + b^2. Here a = x and b = 7, so (x-7)^2 = x^2 - 2(x)(7) + 7^2 = x^2 - 14x + 49. In an area model, the x^2 and 49 corner squares are positive areas, while the two mixed regions contribute a negative middle term because the expression is a difference. Option A includes all three required parts. Option B has a positive middle term and would be the expansion of (x+7)^2. Option C omits the middle term completely and is the difference-of-squares expression x^2 - 49, not a square of a difference. Option D uses only 7x instead of twice 7x. Memory cue: for (a-b)^2, keep the last term positive but make the middle term negative and double the product.

The correct answer is A: the difference-of-squares model. A large square with side x has area x^2. Removing a smaller square with side y removes area y^2, leaving x^2 - y^2. The remaining L-shaped region can be cut and rearranged into a rectangle whose sides are x+y and x-y. Its area is therefore (x+y)(x-y), which proves x^2-y^2 = (x+y)(x-y). Option A describes exactly this removal and rearrangement. Option B represents (x+y)^2, whose expansion contains the positive middle term 2xy. Option C represents (x-y)^2, whose expansion contains the negative middle term -2xy; it is not simply x^2-y^2. Option D concerns boundary length, not the area identity being shown. A useful warning is that difference of squares has no middle term, while squares of sums or differences do have one. Memory cue: remove one square from another, then think difference of squares.

The correct answer is A: both calculations describe the area of the same square. From the outside, the square has side c+d, so its area is (c+d)^2. From the inside, the same square is partitioned into smaller regions. Adding the areas of those regions must give the area of the whole figure because the parts fill the square without gaps or overlaps. This is the basic principle behind a visual proof of an algebraic identity. Option A is correct even though the two calculations look different: they measure the same region in two ways. Option B is false because the inner parts are not a separate unrelated figure; they are parts of the original square. Option C confuses perimeter, a boundary length, with area, a measure of surface. Option D is false because the inner parts occupy real space and have nonzero areas. Memory cue: same complete figure, different partition, same total area.

Direct answer: Option A, ab, is correct. A square with total side a + b can be divided in both directions into lengths a and b. Each middle rectangle has one side of length a and the other side of length b. Using rectangle area = length × breadth, each area is a × b = ab. The two equal rectangles together contribute 2ab to the full square area, giving (a + b)² = a² + 2ab + b². Option B, a², is the square region formed by the two a-length parts. Option C, b², is the other corner square. Option D is a length, not an area, and does not multiply the two dimensions. A common confusion is to report the sum of the side parts instead of their product. For a rectangle, always multiply the perpendicular side lengths.

Direct answer: Option A is correct. Since each side is divided into lengths x and y, the complete side of the large square is x + y. Its total area is therefore (x + y)². The four regions have areas x², xy, xy, and y²; adding them gives x² + xy + xy + y² = x² + 2xy + y². Thus (x + y)² = x² + 2xy + y². Option B has a negative middle term and would require subtractive regions or a side x − y. Option C is the difference-of-squares identity and describes a different arrangement. Option D incorrectly removes the two xy regions; in fact, x² + y² is smaller than (x + y)² by 2xy for positive lengths. Memory cue: two equal positive cross-rectangles create the +2xy term.