Visual models of identities

When the side of the square is divided into lengths \(a\) and \(b\), two rectangles are formed. Each rectangle has area \(a\times b=ab\), so together they represent \(ab+ab=2ab\). The squares with sides \(a\) and \(b\) represent \(a^2\) and \(b^2\), respectively. Exam tip: in the area model for \((a+b)^2\), count both \(ab\) rectangles.

Both sides are equal, so it is a square model of side (x+5). Exam tip: identify a square from equal sides.

The expression begins with an area of x². Removing two strips, each represented by 4x, removes a total of 8x. Adding a small square of area 16 completes the pattern. This is the visual form of a square identity: the first square has side x, the removed strips reduce each relevant side by 4, and the added 16 fills the missing corner. Therefore, the completed square has side length x − 4.

Using the identity \((a-b)^2=a^2-2ab+b^2\), take a=x and b=4. Then \((x-4)^2=x^2-2(x)(4)+4^2=x^2-8x+16\). This exactly matches the described operations: start with x², subtract 8x, and add 16. Hence option B is correct. Option A would produce positive 8x, not the removal described.

Dividing the rectangle according to its side lengths produces four smaller areas: \(x\cdot x=x^2\), \(x\cdot4=4x\), \(3\cdot x=3x\), and \(3\cdot4=12\). Therefore, the two separate parts containing \(x\) are \(3x\) and \(4x\). \(7x\) is the sum of these two parts, not an individual small region. Exam tip: in an area model, multiply the two side lengths of every small rectangle.

It is the area of the whole square with side (a+b). Exam tip: match the sum of parts with the outer side.

The expression \(x^2+2xy+y^2\) matches the identity \((u+v)^2=u^2+2uv+v^2\) when \(u=x\) and \(v=y\). It can therefore be written as \((x+y)^2\). Since the square of a quantity represents the area of a square, the outer side of the area model is \(x+y\).

The model contains one square of area \(x^2\), two rectangles each of area \(xy\), and one square of area \(y^2\). Their total is exactly \(x^2+2xy+y^2\). Hence option B is correct. The expression \((x-y)^2\) would have a negative middle term, while \((2x+y)^2\) and \((x+2y)^2\) would have different first or last terms. The two positive \(xy\) rectangles are a useful visual clue.

The full side of the square is x+y, so its area is \((x+y)^2\). Adding the four regions gives x²+xy+xy+y² = x²+2xy+y². Option B has a negative middle term. Exam tip: combine the two equal xy regions as 2xy.

The area of a square equals the square of its side. Therefore, for side \((g-1)\), the area is \((g-1)^2\). Applying \((a-b)^2=a^2-2ab+b^2\) with \(a=g\) and \(b=1\) gives \(g^2-2g+1\). The expression \(g^2-1\) is a difference of squares, not the square of a difference. Exam tip: the middle term in \((a-b)^2\) is always \(-2ab\).

The large square part has side (2a), so its area is (4a^2). Exam tip: square the whole term (2a).

In a square of side (x+10), the small corner area is (10^2=100). Exam tip: identify the model from the outer side.

Since (9=3^2), the side of the removed small square is (3). Exam tip: take the square root of the removed area.

Since (25=5^2), the total middle term is (2\cdot x\cdot 5=10x). Exam tip: add both equal rectangles.

The area of a rectangle is the product of its side lengths: \((h+6)(h-6)\). Applying \((a+b)(a-b)=a^2-b^2\) gives \(h^2-6^2=h^2-36\). Therefore, option A is correct. \(h^2+36\) incorrectly uses a sum instead of a difference of squares. Exam tip: the product of two matching binomials with opposite signs is always a difference of squares.

The total middle term is (20x), so each rectangle is (10x). Exam tip: divide the total middle term into two equal parts.

The governing idea is the rectangle or factor model for a quadratic expression. In x²+13x+40, the two smaller side lengths must multiply to the constant term 40, while their sum must produce the coefficient 13 of x. The pair 5 and 8 satisfies both relationships: 5×8=40 and 5+8=13. In the visual model, the product 5×8 represents the area of the small corner rectangle, and the two side contributions together create the 13x part. Option B is incorrect because 4×10 equals 40, not 13. Options C and D do not form the required factor pair. Therefore, option A is correct.

A square with side a+b can be divided into four parts: a square of side a, a square of side b, and two rectangles. The first small square has both length and breadth equal to a, so its area is a\times a=a^2. The two rectangles each have area ab, while the second small square has area b^2.

Thus the a^2 part comes specifically from the square of side a, making option A correct. The two ab rectangles produce the middle term 2ab, and the square of side b produces b^2. The perimeter does not directly provide any of these area terms, so option D is not appropriate.

The area of the square is \((x+2y)^2\). Using \((a+b)^2=a^2+2ab+b^2\), with \(a=x\) and \(b=2y\), the combined middle term is \(2\cdot x\cdot 2y=4xy\). \(2xy\) incorrectly ignores the coefficient 2 in \(2y\). Exam tip: include the complete coefficient of each term while finding \(2ab\).

In ((x-7)^2), the middle term is (-14x) and the last term is (49). Exam tip: remember (2\cdot 7).

(x^2-y^2) can be cut and rearranged into a ((x+y)(x-y)) rectangle. Exam tip: observe the removed square.

Reading the same square from outside and from parts gives equal area. Exam tip: this is the main idea of visual proof.