Visual models of identities

In \((x-2)^2=x^2-4x+4\), \(x^2\) and \(4\) are positive. Exam tip: square terms are positive.

In the area model of \((a+b)^2\), each side is divided into lengths \(a\) and \(b\). This produces squares of areas \(a^2\) and \(b^2\), along with two equal rectangles, each of area \(ab\). Their total area is \(ab+ab=2ab\). In \((a-b)^2\), the middle term is \(-2ab\), so it is not the correct model. Exam tip: two equal \(ab\) rectangles in a square identity indicate the middle term \(2ab\).

Two parts in length and two parts in breadth make \(2 \times 2=4\) parts. Exam tip: drawing a grid model is easy.

In the rectangle model, the four parts are \(x^2\), \(4x\), \(x\), and \(4\). The \(x\)-terms are \(4x\) and \(x\), so their total is \(4x+x=5x\). \(4x\) represents only one part, not the sum of both \(x\)-terms. Exam tip: after multiplying binomials, combine like terms with the same variable and exponent.

The constant part comes from \(3 \times 2=6\). Exam tip: multiply the number parts separately.

In the visual model of \((a+b+c)^2\), two rectangles have side lengths \(a\) and \(b\): one represents \(ab\) and the other \(ba\). Since \(ab=ba\), together they give \(2ab\); hence \(ab\) appears twice. The square regions \(a^2\), \(b^2\), and \(c^2\) each occur only once. Exam tip: in the square of a three-term expression, every mixed term has coefficient 2.

There are three square terms \(a^2\), \(b^2\), and \(c^2\). Exam tip: identify square terms and rectangle terms separately.

There are \(3\) parts in each direction so \(3 \times 3=9\) sections form. Exam tip: multiply rows and columns.

Using the identity \((a+b)^2=a^2+2ab+b^2\), the required figure is a square of side \(a+b\). Its area can be divided into areas \(a^2\), two parts of area \(ab\), and \(b^2\). A square of side \(a-b\) gives \((a-b)^2=a^2-2ab+b^2\), so it is not correct. Exam tip: a middle term of \(+2ab\) indicates \((a+b)^2\).

The form \(a^2-2ab+b^2\) is \((a-b)^2\). Exam tip: identify the subtraction square from the negative sign.

Here \(2ab=2 \times 10 \times 3=60\). Exam tip: the middle part comes from two equal rectangles.

The small square has side \(5\), so its area is \(5^2=25\). Exam tip: identify the small square separately.

In the identity \((a-b)^2=a^2-2ab+b^2\), the middle term \(2ab\) is subtracted. Here, \(a=20\) and \(b=2\), so the subtracted term is \(2 \times 20 \times 2\). Although \(2^2\) also appears in the expression, it is added, not subtracted. Exam tip: In \((a-b)^2\), the middle term always has a negative sign.

In the area model for \((x+3)(x+6)\), the four smaller areas are \(x^2\), \(6x\), \(3x\), and \(18\). Their sum is \(x^2+6x+3x+18=x^2+9x+18\). In option A, the middle terms do add to \(9x\), but the constant term is \(14\), not \(18\). Exam tip: for \((x+a)(x+b)\), \(a+b\) gives the coefficient of the middle term and \(ab\) gives the constant term.

In the rectangle model, both the length and breadth are \(x+4\), so its area is \((x+4)(x+4)\). This is exactly the area of a square with side \(x+4\), written as \((x+4)^2\). Therefore, both models represent the same area. Option A is incorrect because the two expressions are equal. Exam tip: Any product of the form \(a\cdot a\) can be written as \(a^2\).

In \(a^2-b^2\), the smaller square area is subtracted from the larger square. Exam tip: distinguish square of difference and difference of squares.

The area of a rectangle equals the product of its side lengths. Here it is \((a+b)(a-b)\). Using the difference-of-squares identity, \((a+b)(a-b)=a^2-b^2\), so \(a^2-b^2\) is correct. \((a-b)^2\) would require both sides to be \(a-b\). Exam tip: whenever you see \((x+y)(x-y)\), use \(x^2-y^2\).

Using the identity \((x+y)^2=x^2+2xy+y^2\), the given expression is the area of a square whose side is \(x+y\). In the square model, regions of areas \(x^2\), \(y^2\), and two \(xy\) regions together form the whole square. A side of \(x-y\) would give \(x^2-2xy+y^2\), so it is not correct. Exam tip: Write the square of the side and match it with the relevant identity.

\(x^2-6x+9=x^2-2\times x\times3+3^2=(x-3)^2\). Therefore, it represents the area of a square with side \(x-3\). A square of side \(x+3\) gives \((x+3)^2=x^2+6x+9\), whose middle term is positive. Exam tip: In \(x^2-2ax+a^2=(x-a)^2\), always check the sign of the middle term.

Here, \(21=20+1\) and \(19=20-1\). Thus, the product has the form \((20+1)(20-1)\), which uses the identity \((a+b)(a-b)=a^2-b^2\). Hence, \(21\times19=20^2-1^2=399\). The identity \((a+b)^2\) is for squaring one binomial, so it does not apply here. Exam tip: when two numbers are equally spaced from a middle number, look for the difference-of-squares identity.