Visual models of identities

Each rectangle has area (5x), so the total is (10x). Exam tip: treat the middle term as the sum of two equal rectangles.

Here (a=7) and (b=3), so the whole area is ((7+3)^2=100). Exam tip: identify the side from the square area.

(49=7^2) and (14x=2\cdot7x). Exam tip: take the square root of the constant term to find the small square side.

The removed-square model gives (p^2-q^2=(p-q)(p+q)). Exam tip: connect difference of squares with rectangle area.

(16=4^2) and (-8x=-2\cdot4x), so the side is (x-4). Exam tip: read the negative middle term as a subtraction model.

The total area of the two removed rectangles is (2\cdot40\cdot2=160). Exam tip: in ((a-b)^2), (2ab) is subtracted.

((x+6)(x-2)=x^2-2x+6x-12=x^2+4x-12). Exam tip: handle positive and negative area signs carefully.

The grid gives (x\cdot x), (x\cdot(-3)), (7\cdot x), and (7\cdot(-3)). Exam tip: multiply each row and column.

(4x^2=(2x)^2), (9=3^2), and the middle term is (2\cdot2x\cdot3=12x). Exam tip: combine the sides from both square terms.

(9a^2=(3a)^2) and (16b^2=(4b)^2); the middle term is negative, so it is ((3a-4b)^2). Exam tip: use the sign to decide addition or subtraction.

When the square is divided into lengths \(a\), \(b\), and \(c\) in both directions, there are two rectangles with area \(ab\): one \(a\times b\) rectangle and one \(b\times a\) rectangle. Their total area is \(ab+ba=2ab\). \(ab\) is the area of only one rectangle, so it is a close but incomplete option. Exam tip: in the expansion of a square, cross-product terms generally occur twice.

The total area of two equal rectangles is (2\cdot2x\cdot5=20x). Exam tip: do not forget the factor (2) in the middle term.

The large square has side \(a+b\). Its partition gives squares of areas \(a^2\) and \(b^2\), plus two rectangles each of area \(a\times b\); together they give \(2ab\). Exam tip: count both rectangular regions.

Split (6x) into two (3x) rectangles, so the corner is (3^2=9). Exam tip: take half of the middle coefficient and square it.

Half of (-10x) is (-5x), and the corner area is (5^2=25). Exam tip: keep the sign separate and take the square as positive.

The two shaded rectangles have area (ab+ab=2ab). Exam tip: count equal rectangles and add them.

The rectangle’s area is \((x+y)(x-y)\). Using the identity, \((x+y)(x-y)=x^2-y^2\), so A is correct. Option D has area \((x+y)^2\). Exam tip: conjugate binomials produce a difference of squares.

Subtracting the two square areas leaves four (ab) regions. Exam tip: identify the remaining parts in a difference model.

The numbers 27 and 33 are equally spaced, by 3, on either side of 30. Using \((a-b)(a+b)=a^2-b^2\), \((30-3)(30+3)=30^2-3^2=900-9=891\). The value 900 is only \(30^2\); it does not account for subtracting \(3^2\). Exam tip: when two numbers are equally distant from a middle number, use the difference-of-squares identity for quick multiplication.

The rectangle’s area is found by multiplying its two side lengths. Its sides are \\(2x+3\\) and \\(2x-3\\), which form a sum-and-difference pair. Using \\((u+v)(u-v)=u^2-v^2\\), let \\(u=2x\\) and \\(v=3\\). The area is therefore \\((2x)^2-3^2\\).

Evaluating the squares gives \\(4x^2-9\\), so option A is correct. Direct multiplication gives the same result: \\(2x(2x-3)+3(2x-3)=4x^2-6x+6x-9=4x^2-9\\). The terms involving \\(x\\) cancel because one side contains plus 3 and the other contains minus 3. Thus option B, which has a plus sign, does not match the identity.