Visual models of identities

Since (64=8^2), the combined middle term is (2\cdot x\cdot 8=16x). Exam tip: get side (8) from the corner square.

In the four parts, (xy) and (-xy) cancel each other. Exam tip: middle terms cancel in the sum-and-difference model.

(m^2+2mn+n^2=(m+n)^2) and the square root of (196) is (14). Exam tip: take the square root of the whole model.

To find the combined term containing one \(x\), multiply each part of the first factor by each part of the second factor. In \((x-6)(x+2)\), the two terms involving one \(x\) come from \(x\cdot 2=2x\) and \((-6)\cdot x=-6x\). They must be combined with their signs, just as positive and negative areas would be combined in a visual rectangle model.

Adding these like terms gives \(2x-6x=-4x\). Thus the combined \(x\)-term is negative and has value \(4x\) in magnitude. Option B is correct. The constant product is \(-12\), but it is not part of the requested \(x\)-term. The sign must not be lost when the positive and negative rectangles are added.

The whole square has side \(x+y\), so its area is \((x+y)^2\). Adding its four regions gives \(x^2+xy+xy+y^2=x^2+2xy+y^2\). In \((x-y)^2\), the middle term is negative. Exam tip: count both \(xy\) rectangles.

(81=9^2) and the middle term is (-18x), so the side is (x-9). Exam tip: get the number from last term and sign from middle term.

The bigger area is (121) and the smaller is (49), so the difference is (72). Exam tip: subtract the square areas first.

Removing two \(ab\) rectangles gives \(a^2-ab-ab\). Their overlap \(b^2\) was subtracted twice, so add it once: \(a^2-2ab+b^2\). Exam tip: removal in an area model indicates a negative term.

(a=8) and (b=6), so (2ab=96). Exam tip: find side lengths from square areas and add rectangle areas.

Since ((x+y)^2-(x-y)^2=4xy), the first square is greater than the second by (4xy). Exam tip: remember the difference of these two squares.

The two equal rectangles have area (ab) and the small square has area (b^2). Exam tip: split the big square into smaller regions before writing the identity.

After subtracting two strips the common corner must be added back. Exam tip: check overlap when subtracting areas.

The total strip area is (7x+3x=10x). Exam tip: in ((x+a)(x+b)), the middle coefficient is (a+b).

Each rectangle has area (mn), so together they give (2mn). Exam tip: do not forget the number of equal rectangles.

Rearranging the difference of squares forms the rectangle ((r-s)(r+s)). Exam tip: identify (r^2-s^2) as a difference of squares.

The four small width parts show an increase of (4) along each side. Exam tip: observe the total added width on the edge.

Here (36) is (6^2) and (12x) is (2\cdot6\cdot x). Exam tip: find the added width from the constant square.

The small square has side (9y) and the two rectangles together give (18xy). Exam tip: the sum of the two rectangles directly gives the middle term.

Each pair forms two equal rectangles, so the total is (2ab+2bc+2ca). Exam tip: count pairwise rectangles twice in a trinomial square.

Because (5+6=11) and (5\cdot6=30). Exam tip: match both sum and product in a rectangle model.