Visual models of identities

The area of the larger square is \((x+4)^2\), and that of the smaller square is \((x-4)^2\). Thus, the difference is \((x+4)^2-(x-4)^2\). Using \((a+b)^2-(a-b)^2=4ab\), with \(a=x\) and \(b=4\), we get \(4\times x\times4=16x\). The expression \(x^2-16\) is the area of the smaller square, not the difference of the areas. Exam tip: subtract the smaller area from the larger area before simplifying.

Dividing a square of side \(a+b\) into lengths \(a\) and \(b\) gives an \(a^2\) square, a \(b^2\) square, and two \(ab\) rectangles. Its area is \(a^2+ab+ab+b^2\). Option B misses one \(ab\) region. Exam tip: always count both equal rectangles.

(81-18x+x^2=x^2-18x+81=(x-9)^2), equal to the square of (9-x). Exam tip: arrange terms in order first.

Since \(36=6^2\), the perfect-square expression must be \((x+6)^2\). Expanding it gives \((x+6)^2=x^2+12x+36\). Therefore, \(a=12\). Here, \(6\) is only the square root of the constant term, not the coefficient of the middle term. Exam tip: in \((x+b)^2=x^2+2bx+b^2\), the coefficient of the middle term is always \(2b\).

(4x^2=(2x)^2) and (49=7^2), so the middle term is (2\cdot2x\cdot7=28x). Exam tip: multiply twice the two sides.

The middle area is (2\cdot2a\cdot3b=12ab). Exam tip: include coefficients in the rectangle sides.

The area of a square is the square of its side. Therefore, \((5a-2b)^2=(5a)^2-2(5a)(2b)+(2b)^2=25a^2-20ab+4b^2\). The minus sign makes the middle term negative. Option B has the wrong sign in the middle term, while option C misses the factor 2. Exam tip: in \((x-y)^2=x^2-2xy+y^2\), always include 2 in the middle term.

Difference of squares comes from the removal model, not from the full addition square. Exam tip: look for the removed part.

The governing identity is (x + a)(x + b) = x² + (a + b)x + ab. Comparing this general expansion with x² + 9x + 20 gives two conditions that must hold at the same time: a + b = 9 and ab = 20. The pair 4 and 5 satisfies both conditions because 4 + 5 = 9 and 4 × 5 = 20. Substitution verifies the result: (x + 4)(x + 5) = x² + 5x + 4x + 20 = x² + 9x + 20. Hence option A is correct. The pair 2 and 10 has the correct product but sum 12; 1 and 20 has sum 21; and 3 and 6 has sum 9 but product 18. Thus no other option matches both coefficients.

This is ((x-6)^2), and the total area of the two removed rectangles is (2\cdot6x=12x). Exam tip: identify the subtracted middle area.

Each rectangle has area (2x\times3=6x), so the total is (12x). Exam tip: the middle term comes from two equal rectangles.

Since (9x^2=(3x)^2) and (4=2^2), the side is (3x+2). Exam tip: use square roots of square terms to identify the outer side.

While removing two strips, the corner (b^2) part is subtracted twice. Exam tip: remember to add back the over-subtracted part.

After removing the smaller square from the larger square, the remaining area rearranges into the rectangle ((a+b)(a-b)). Exam tip: link difference of squares with the rectangle model.

One rectangle is (2xy) and the other is (-2xy), so the middle terms cancel. Exam tip: identify cancellation in conjugate products.

The total middle term is (10x), so each equal rectangle has area (5x). Exam tip: split the middle term into two equal parts.

The area of a square is found by multiplying its side by itself. For side \(3p-4\), the area is \((3p-4)^2\). Using \((a-b)^2=a^2-2ab+b^2\), take \(a=3p\) and \(b=4\). The first part is \((3p)^2=9p^2\), the middle part is \(-2(3p)(4)=-24p\), and the last part is \(4^2=16\).

Combining these parts gives \(9p^2-24p+16\), so option A is correct. The negative middle term appears because the side contains subtraction. Option B misses the factor 2 in the middle term, option C does not square the coefficient 3 correctly, and option D changes the sign. The area model represents the same three parts: one square, two negative rectangles, and one small square.

Since (16x^2=(4x)^2) and (25=5^2), the dimensions are (4x+5) and (4x-5). Exam tip: first find the side of both squares.

The small square gives (a^2=49), so (a=7), and (2ax=14x) confirms it. Exam tip: match the answer with both clues.

The rectangle parts come from (x\times x), (x\times8), (3\times x), and (3\times8). Exam tip: multiply each outside part.