Visual models of identities

In the square model of \((a+b)^2\), two equal rectangles are \(ab\). Exam tip: write the total middle area as \(2ab\).

Splitting the length into \(r\) and \(s\) gives area \(r^2+rs\). Exam tip: connect distributive law with area.

One \(x^2\) tile represents area \(x^2\), seven \(x\)-tiles represent a total area of \(7x\), and 12 unit tiles represent 12. Hence, the total area is \(x^2+7x+12\). The expression \(x^2-7x+12\) would require negative \(x\)-tiles, which are not stated in the question. Exam tip: write the value of each type of tile separately and then add like terms.

The four sub-rectangles have areas \(x\cdot x=x^2\), \(x\cdot 3=3x\), \(2\cdot x=2x\), and \(2\cdot 3=6\). Their sum is \(x^2+3x+2x+6=x^2+5x+6\), so option C is correct. In option B, the middle term \(6x\) is incorrect. Exam tip: in a rectangle model, multiply the side lengths of each small part and then add all the areas.

The square has side \(t+1\), so its area is \((t+1)^2\). In the area model, the \(t\times t\) part gives \(t^2\), the two \(t\times1\) rectangles together give \(2t\), and the \(1\times1\) part gives \(1\). Therefore, the result is \(t^2+2t+1\). Option \(t^2+1\) misses the areas of the two rectangles. Exam tip: use \((a+b)^2=a^2+2ab+b^2\).

This is the visual explanation of difference of squares. Exam tip: remember the rearrangement as \((a+b)(a-b)\).

\((10-3)^2\) equals \(10^2-2\times10\times3+3^2\). Exam tip: do not forget the subtraction of \(2ab\).

Such a model has squares like \(a^2\), \(b^2\), \(c^2\) and rectangles like \(ab\), \(bc\), \(ca\). Exam tip: identify area terms by shape.

With \(4x\) and \(2x\), the corner \(8\) shows the numbers are \(4\) and \(2\). Exam tip: look at the product in the corner.

Two squares and two rectangles form one complete square. Exam tip: identify a perfect square trinomial from the figure.

The area of a square equals the square of its side. Exam tip: think of a square for \((p+q)^2\).

The total area is the sum of the areas of all parts: \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Therefore, option C is correct. Option A leaves out the areas of the two \(ab\) rectangles. Exam tip: in visual-area questions, list the area of each smaller region before adding them.

In \((z-1)^2=z^2-2z+1\), the last square term \(1\) is positive. Exam tip: keep the squared term positive.

The total area is \(a^2+ab\), which can be written as \(a(a+b)\). Exam tip: identify the common factor from the diagram.

The sum of parts is \(x^2+6x+5\), which matches \((x+5)(x+1)\). Exam tip: read the outside dimensions.

In an area model, the same figure's area is written in different forms. Exam tip: keep both forms equal.

The square model has two equal rectangles made from \(x\) and \(y\). Exam tip: convert them into \(2xy\).

When both dimensions are equal, the figure is a square. Exam tip: treat equal length and breadth as a square.

Two \(ab\) strips are removed, so the total subtraction is \(2ab\). Exam tip: \(b^2\) is added back later.

The area of a square is the square of its side. Using \((a+b)^2=a^2+2ab+b^2\), with \(a=7\) and \(b=3\), we get \((7+3)^2=7^2+2\times7\times3+3^2=49+42+9=100\). \(49\) is only \(7^2\), so it misses both \(3\) and the middle term \(2\times7\times3\). Exam tip: for a square whose side is a sum, include all three terms: \(a^2\), \(2ab\), and \(b^2\).