Visual models of identities

Correct answer: A, a square with side \(a+b\). The area of that square is \((a+b)^2\). Divide each side into parts \(a\) and \(b\): the four regions have areas \(a^2\), \(ab\), \(ab\), and \(b^2\). Their sum is \(a^2+2ab+b^2\), so both expressions describe the same total area. Option B gives \((a-b)^2=a^2-2ab+b^2\), with a negative middle term. Option C gives \((a+b)(a-b)=a^2-b^2\), the difference of squares. Option D has area only \(ab\), so it cannot equal the complete expression in general. Memory cue: the outside shape tells you the squared binomial; a side marked \(a+b\) means the positive-square identity.

Correct answer: B, \(x^2-14x+49\). The square’s area is \((x-7)^2\). Apply \((p-q)^2=p^2-2pq+q^2\) with \(p=x\) and \(q=7\): \(x^2-2(x)(7)+7^2=x^2-14x+49\). In a visual model, the large \(x\times x\) square contributes \(x^2\), two strips of area \(7x\) are removed, and the \(7\times7\) corner is added back, giving \(-14x+49\). Option A subtracts only one strip. Option C is \((x+7)(x-7)\), a difference of squares, not \((x-7)^2\). Option D has the sign for \((x+7)^2\). Memory cue: squaring a difference creates a negative double-product term, not merely \(-7x\).

The governing concept is algebraic comparison of equivalent area expressions. The rectangle has dimensions x + 4 and x + 6, so its area is (x+4)(x+6) = x² + 10x + 24. The square has side x + 5, so its area is (x+5)² = x² + 10x + 25. Subtracting the rectangle’s area from the square’s area gives (x²+10x+25) − (x²+10x+24) = 1. Thus the square is larger by exactly one square unit whenever the stated dimensions are valid lengths. This also illustrates that numbers equally spaced around x+5 produce a slightly smaller product than the square of their average. Therefore option B is correct. The x² and 10x terms cancel, so options C and D cannot be correct.

Direct answer: Option B, x + 8, is correct. In a square area model, the complete expression must be the square of the side length. Compare the expression with the identity (p + q)² = p² + 2pq + q². Here p = x and q = 8, because x² is the first square and 64 = 8² is the corner square. The middle term is 2 × x × 8 = 16x, exactly as given. Therefore x² + 16x + 64 = (x + 8)², so the outer side is x + 8. Option A incorrectly treats the middle coefficient 16 as a side. Option C would produce (x − 8)² = x² − 16x + 64, with the wrong sign. Option D uses 64, which is an area, not a side length. Memory cue: the middle coefficient is twice the constant part of the side, not that part itself; always expand the proposed side to check.

Direct answer: Option A, (a + b)², is correct. Imagine that each side of the large square is split into lengths a and b. The complete side is therefore a + b, so the total area is (a + b)². The four regions have areas a² for the first square, ab and ab for the two equal rectangles, and b² for the smaller square. Adding them gives a² + ab + ab + b² = a² + 2ab + b², which is exactly (a + b)². Option B represents a subtraction arrangement and has a negative middle term, −2ab. Option C is the difference of two squares and normally represents a rectangle, not this four-part square. Option D omits both rectangular regions. A useful cue is that two positive equal rectangles always contribute +2ab.

Direct answer: Option A, 2x² + 11x + 12, is correct. The area of a rectangle is length multiplied by breadth, so calculate (2x + 3)(x + 4). Use a four-cell area model: 2x × x = 2x², 2x × 4 = 8x, 3 × x = 3x, and 3 × 4 = 12. Now combine like terms: 2x² + 8x + 3x + 12 = 2x² + 11x + 12. Option B includes only 7x and misses the 8x contribution. Option C has the wrong constant and combines the terms incorrectly. Option D reverses the correct middle and constant information. A reliable check is to multiply each term in the first bracket by every term in the second bracket, then combine only like terms.

Direct answer: Option A, 4 and −12x, is correct. Split the side 3x − 2 into the two parts 3x and −2. In the square model, the small corner is made from the constant part, so its area is (−2)² = 4. The two middle rectangles each have area (3x)(−2) = −6x. Together they contribute −6x + −6x = −12x. This also follows from (3x − 2)² = 9x² − 12x + 4. Option B gives half of each relevant result and therefore misses the full contribution. Option C has the correct corner but loses the negative sign. Option D gives 9x², which is the large x-part square, not the small corner. Remember that squaring −2 makes a positive corner area, but multiplying 3x by −2 keeps the negative sign in the middle terms.

Direct answer: Option A, 4x + 8, is correct. Expand both model areas separately. The first is (x + 2)² = x² + 4x + 4. The second uses the difference-of-squares identity: (x + 2)(x − 2) = x² − 4. Subtract the second area from the first: (x² + 4x + 4) − (x² − 4) = x² + 4x + 4 − x² + 4 = 4x + 8. Option B gives only the difference between the constant parts and ignores the 4x strip. Option C ignores the variable contribution and is incomplete. Option D represents only part of the middle contribution. The key warning is to keep the minus sign when subtracting the second expression; it changes −4 into +4 during subtraction. Expanding both areas before comparing prevents confusion.

Direct answer: Option C, 6, is correct. A square with side x + 6 has area (x + 6)² = x² + 12x + 36. The visual model contains two equal middle rectangles, each labelled kx, so together they contribute 2kx. Comparing the middle terms gives 2kx = 12x. For x not treated as the zero variable in the coefficient comparison, 2k = 12, so k = 6. Option A is 12, the total coefficient of both middle rectangles, not the coefficient of one rectangle. Option B is 36, the area of the small corner square, and option D is not supported by the expansion. The important distinction is between each strip, kx, and the pair of strips, 2kx. A quick check is x² + 6x + 6x + 36 = x² + 12x + 36.

Direct answer: Option B, 4ab, is correct. Divide each side of the square into parts a and 2b. The two mixed rectangular regions have dimensions a by 2b. Therefore each one has area a × 2b = 2ab. Since there are two such rectangles, their combined area is 2ab + 2ab = 4ab. This is also the mixed term in the expansion (a + 2b)² = a² + 4ab + 4b². Option A counts only one rectangle and misses its equal partner. Option C is the area of the square whose side is 2b, not a mixed rectangle. Option D adds the two square regions a² and 4b² and excludes the rectangles. Memory cue: the cross term is always twice the product of the two side parts: 2 × a × 2b = 4ab.

The correct answer is 4. To compare the two expressions, subtract the first expression from the second: (x² + 6x + 9) − (x² + 6x + 5). Remove the brackets carefully: x² + 6x + 9 − x² − 6x − 5. The x² terms cancel, and the 6x terms also cancel because they are common to both models. Only the constants remain, so 9 − 5 = 4. Therefore, the second model has four more square units than the first, provided the visual model represents these algebraic areas. Option A is wrong because 6x is a common middle part, not the increase. Option B is only the first constant part. Option D is only the second constant part, not the difference. A useful memory cue is: when two expressions have the same variable terms, compare only the terms that changed.

The correct answer is a square with side 3x + 5. Use the identity (a + b)² = a² + 2ab + b². In the given expression, 9x² is (3x)² and 25 is 5². The middle term predicted by the identity is 2(3x)(5) = 30x, which is exactly the given middle term. Therefore, 9x² + 30x + 25 = (3x + 5)², so a square whose side is 3x + 5 represents the expression. Option B is incorrect because squaring 9x + 25 would produce much higher-degree and different terms. Option C gives (3x − 5)² = 9x² − 30x + 25, with a negative middle term. Option D also gives the wrong expansion. Check the first and last terms by taking their square roots, then verify the middle term by doubling their product.

The correct answer is 12x + 36. The area of a square is the square of its side. Thus, the larger square has area (x + 6)², while the removed square has area x². The remaining area is (x + 6)² − x². Expanding the first square gives x² + 12x + 36. Subtracting x² leaves x² + 12x + 36 − x² = 12x + 36. Option A is therefore correct. Option B uses 6x instead of the required middle term 2 × x × 6 = 12x. Option C fails to subtract the x² area of the smaller square. Option D has the correct 12x part but an incorrect constant term; the square of 6 is 36, not 6. The difference-of-squares identity gives a quicker check: (x + 6)² − x² = 6(2x + 6) = 12x + 36.

The correct answer is x² + 6x − 27. The area of a rectangle equals length multiplied by breadth, so calculate (x − 3)(x + 9). Multiply every term in the first bracket by every term in the second: x·x = x², x·9 = 9x, (−3)·x = −3x, and (−3)·9 = −27. Combining like terms gives 9x − 3x = 6x, so the area is x² + 6x − 27. Option B has the wrong middle term and sign. Option C incorrectly adds 9x and 3x instead of subtracting 3x. Option D changes the negative product (−3)(9) into a positive 27. The negative constant is not an error; it follows from signed multiplication. A reliable method is to use a four-cell area grid and write the sign of every partial product before combining terms.

The correct model removes the square of side y from the square of side x + y. The expression itself is a subtraction of two areas: the large square has area (x + y)², and the removed small square has area y². The remaining area is (x + y)² − y². Using the difference-of-squares identity, this becomes [(x + y) − y][(x + y) + y] = x(x + 2y). Option A matches the original subtraction exactly. Option B represents (x − y)², which is a different expression and includes a different middle term. Option C represents only the product xy, not the full remaining area x(x + 2y). Option D keeps the removed part, so it represents only (x + y)² and does not perform the subtraction. The key visual idea is that subtraction of algebraic expressions corresponds to removing one area from another.

The correct sides are 2x + 5 and x + 2. Multiply the proposed factors: (2x + 5)(x + 2). The four partial products are 2x·x = 2x², 2x·2 = 4x, 5·x = 5x, and 5·2 = 10. Adding the middle terms gives 4x + 5x = 9x, so the product is 2x² + 9x + 10, exactly the stated rectangle area. Option B produces 2x² + 21x + 10, not the target. Option C produces x² + 9x + 20, so even its first and last terms are wrong. Option D produces 2x² + 12x + 10, whose middle term is 12x. A useful check for factor pairs is to verify the leading term, the constant term, and then the sum of the cross-products. All three checks must agree before accepting the sides.

The correct answer is 4x. The square with side x + 1 has the larger area, so subtract the smaller area from it: (x + 1)² − (x − 1)². Expand both squares: (x + 1)² = x² + 2x + 1 and (x − 1)² = x² − 2x + 1. Their difference is x² + 2x + 1 − x² + 2x − 1 = 4x. The same result follows quickly from a² − b² = (a − b)(a + b): here the difference is [(x + 1) − (x − 1)][(x + 1) + (x − 1)] = 2 × 2x = 4x. Option A misses one of the two cross-effects. Option C is only the product of the differences, not the area difference. Option D is (x − 1)(x + 1), not the difference of their squares. Always subtract the smaller area from the larger one.

The correct answer is two rectangles, each with area xy. The identity (x + y)² = x² + 2xy + y² can be represented by a large square divided into four parts: one square of area x², one square of area y², and two rectangles. Each rectangle has side lengths x and y, so each has area xy. Together, the two rectangles contribute xy + xy = 2xy, which explains the middle term. Option A lists side lengths rather than the areas of the remaining regions. Option B incorrectly says each rectangle has area 2xy; then their total would be 4xy. Option D repeats the two square regions and does not account for the middle term. A useful memory cue is that the middle term in a square identity is made from two equal cross-rectangles, not one large rectangle.