Visual models of identities

The area of the large square is \(z^2\), while the removed square has area \(5^2=25\). Thus, the remaining area is \(z^2-25\). Using the difference of squares identity, \(z^2-25=(z-5)(z+5)\). Hence, the rectangle can have sides \(z-5\) and \(z+5\). The sides \(z-5\) and \(z-5\) would give \((z-5)^2\), which is not equal to \(z^2-25\). Exam tip: recognise \(a^2-b^2=(a-b)(a+b)\) before expanding expressions.

The parts are (x^2) (2x) (7x) and (14) so the total is (x^2+9x+14). Exam tip: add the cross rectangle areas.

The area of a rectangle is the product of its side lengths: \((t+3)(t-4)\). Using distribution gives \(t^2-4t+3t-12=t^2-t-12\). Therefore, option B is correct. Option A incorrectly adds the middle terms to get \(7t\). Exam tip: while multiplying binomials, write all four products first and then combine like terms.

(36=6^2) and (12k=2\cdot k\cdot 6) so the side is (k+6). Exam tip: find the small square side first.

The two rectangles have areas (ab) and (ab) so the contribution is (2ab). Exam tip: the middle term comes from this part.

The total area of two rectangles is (2\cdot 7\cdot 3=42). Exam tip: connect the numerical model with (2ab).

The remaining area is (12^2-5^2=144-25=119). Exam tip: subtract the small square area from the large square area.

The area of a square is the square of its side. Therefore, for side \((n-2)\), the area is \((n-2)^2\). Using \((a-b)^2=a^2-2ab+b^2\), with \(a=n\) and \(b=2\), we get \(n^2-2(n)(2)+2^2=n^2-4n+4\). The expression \(n^2-4\) is not the square of a difference; it equals \((n-2)(n+2)\). Exam tip: in \((a-b)^2\), the middle term is always negative, \(-2ab\).

(5\cdot 6=30) and (5+6=11) so the sides are (x+5) and (x+6). Exam tip: match both product and sum.

(49=7^2) and (14s=2\cdot s\cdot 7) so the small part is (7). Exam tip: take the square root of the constant term.

((2x+3)^2=(2x)^2+2\cdot 2x\cdot 3+3^2=4x^2+12x+9). Exam tip: treat (2x) as the whole first part.

The total area is \((3a-2)(3a-2)=(3a-2)^2\). Using \((x-y)^2=x^2-2xy+y^2\), with \(x=3a\) and \(y=2\), gives \(9a^2-12a+4\). Option D has a positive middle term, which would occur for \((3a+2)^2\). Exam tip: in the square of a difference, the middle term is negative.

(4x^2=(2x)^2) and (9=3^2) so the form is ((2x-3)(2x+3)). Exam tip: rewrite both terms as squares.

The small square has side (4) so its area is (4^2=16). Exam tip: identify the small square from strip width.

In difference of squares the remaining parts rearrange into a rectangle ((a-b)(a+b)). Exam tip: remember the rearrangement model.

This is (x^2+2\cdot x\cdot 5+5^2) so it is ((x+5)^2). Exam tip: view (10x) as two equal strips.

The area of a rectangle is the product of its side lengths: \((x+9)(x-1)=x^2-x+9x-9=x^2+8x-9\). Therefore, the correct area is \(x^2+8x-9\). In option A, the middle terms have been combined with incorrect signs. Exam tip: after multiplying, carefully combine like terms such as \(-x\) and \(9x\).

Two equal rectangles are formed for each pair so (2ab+2bc+2ca) appears. Exam tip: count pairwise rectangles in a three-part square.

((p+q)^2) has two (pq) rectangles so one (pq) rectangle is missing. Exam tip: remember the middle term (2pq).

Each strip has sides (x) and (6) so its area is (6x). Exam tip: write strip area as (x\cdot 6).