Visual models of identities

When two strips are removed, their common corner square \(b^2\) is subtracted twice. Add it once back: \(a^2-2ab+b^2=(a-b)^2\). Exam tip: always check overlapping regions in area models.

This visual model shows (a^2-2ab+b^2). Remember the overlapped removed part is added back once.

The area ((a+b)(a-b)) equals (a^2-b^2). In exams identify binomials with opposite signs.

Two (4x) strips combine to give (8x). In exams combine equal rectangular strips to form the middle term.

The small corner has side (6), so its area is (36). In exams identify the square of the constant part separately.

The area is (y^2+10y+25), which is ((y+5)^2). In exams halve the middle coefficient to identify the constant side.

In ((m-3)^2=m^2-6m+9), the middle term is (-6m). In exams negative strips show subtraction.

The two strips give (7x+7x=14x) and the corner gives (49). In exams check how many strips are added.

A square with side (r+2) has two (2r) strips and a (4) corner. In exams read the square model as four basic parts.

The parts give area (x^2+8x+15), so the sides are (x+3) and (x+5). In exams the small rectangle gives the constant product.

Since (4+5=9) and (4\cdot5=20), the sides are (x+4) and (x+5). In exams match the sum with the middle term and the product with the constant term.

((a+4)(a-4)=a^2-16). In exams connect difference of squares with a rectangle having sum and difference as sides.

Since (8=10-2), this is a model of ((a-b)^2). In exams do not forget to add back the overlap in a reduced square.

Using (99^2=(100-1)^2) gives (10000-200+1). In exams using the nearest round number makes the visual model easier.

The middle parts are (2x\cdot3) and (3\cdot2x), so the total is (12x). In exams put (a=2x) and (b=3) in (2ab).

In the area model for side \((3y+2)\), the side is split into \(3y\) and \(2\). The product \(3y\times3y=9y^2\) forms the largest square-type region. \(6y\) is the combined area of two rectangles, while \(4\) is the smaller \(2\times2\) square. Exam tip: identify the main squared term by multiplying the two variable-side parts first.

Here (4x^2=(2x)^2) and (9=3^2), and the middle term is (2\cdot2x\cdot3=12x). In exams add the sides of the two squares.

The two middle parts (-10a) and (-10a) combine to (-20a). In exams read negative signs in subtraction models carefully.

The strips are (2x) and (7x), whose sum is (9x). In exams the middle term in ((x+a)(x+b)) is ((a+b)x).

The strips (6x) and (4x) show the sides (x+6) and (x+4). In exams the strips directly give the two constant parts.