Visual models of identities

The expression \(12^2-2\times12\times2+2^2\) matches the identity \(a^2-2ab+b^2=(a-b)^2\), with \(a=12\) and \(b=2\). Therefore, it equals \((12-2)^2\), the area of a square with side \(12-2\). A side of \(12+2\) would give \((a+b)^2\), whose middle term is positive. Exam tip: when the middle term is negative, check for the identity \((a-b)^2\).

In difference of squares, dimensions become \((a+b)\) and \((a-b)\). Exam tip: compute \(9^2-4^2=(13)(5)\).

Splitting the rectangle into parts with widths \(x\) and \(6\) gives \(x^2+6x\). Exam tip: see distribution as area.

There is one \(ab\) part above and one \(ab\) part beside. Exam tip: add both as \(2ab\).

These tiles form a square of side \(x+y\). Exam tip: identify the outer side of a perfect square.

Two \(2x\) rectangles make \(4x\), and \(2^2=4\). Exam tip: check the side of the small square.

An area model shows multiplication and addition in a diagram. Exam tip: take area as length times breadth.

The small square has side \(4\), so its area is \(4^2=16\). Exam tip: square the constant number.

The constant corner is formed by \(2\times5=10\). Exam tip: connect the final constant term with the corner area.

Two rectangular parts \(x\) and \(4x\) combine to make \(5x\). Exam tip: add the \(x\)-terms.

The square area is \((u-v)^2\), which becomes \(u^2-2uv+v^2\). Exam tip: keep the middle term negative.

\(15^2-5^2=(15+5)(15-5)=20\times10=200\). Exam tip: calculate quickly using difference of squares.

\(8x\) splits into two equal \(4x\) rectangles, so \(4^2=16\) is added. Exam tip: square half the coefficient.

A square divided into three parts creates two equal rectangles for each pair. Exam tip: double the paired rectangles.

The area of a square is the square of its side. Here the side is \(20-1\), so using \((a-b)^2=a^2-2ab+b^2\), we get \((20-1)^2=20^2-2\times20\times1+1^2=400-40+1=361\). Therefore, \(361\) is correct. \(399\) results from subtracting only 1 from 400, but the middle term \(-2ab\) must also be included. Exam tip: always write the middle term when expanding \((a-b)^2\).

In the rectangle model of \(a^2-b^2\), the middle \(2ab\) term does not appear. Exam tip: connect each identity with its own model.

The two rectangles have areas \(6x\) and \(6x\), so the total is \(12x\). Exam tip: double equal rectangles.

\(49=7^2\) and \(14x=2\times7\times x\). Exam tip: identify the side from the square root of the last term.

In \((a+b)^2\), all parts form a large square without gaps. Exam tip: read the outer side first.

A visual model helps us understand the area of each term. Exam tip: verify the formula through the diagram.