Visual models of identities

Each pair has two rectangles, so the total is (2ab+2bc+2ca). In exams, count each mixed pair twice.

Dividing the square gives \(x\times x=x^2\), \(y\times y=y^2\), and two rectangles of \(x\times y=xy\). Thus the tiles represent \(x^2+2xy+y^2\). Option A misses one \(xy\) tile. In exams, count all four regions of the model.

The four signed areas are \(x\cdot x=x^2\), \(x\cdot(-5)=-5x\), \(3\cdot x=3x\), and \(3\cdot(-5)=-15\). Adding them gives \(x^2-5x+3x-15=x^2-2x-15\). Option B incorrectly adds the two middle terms. Exam tip: multiply the constant terms for the constant term and check its sign carefully.

In a signed rectangle, the constant corner is obtained by multiplying the constant terms of the two binomials: \((-2)\times(-9)=18\). Therefore, the constant corner is 18.

The value -18 would arise if one constant term were positive and the other negative. Exam tip: the product of two negative numbers is positive.

Dividing a square of side \(a+b\) gives one \(a^2\) square, one \(b^2\) square, and two \(a \times b\) rectangles. Thus its area is \(a^2+2ab+b^2\). Exam tip: count both identical middle rectangles.

The side of the whole square is \(a+b\), so its area is \((a+b)^2\). The partition forms squares of areas \(a^2\) and \(b^2\), along with two rectangles, each of area \(ab\). Hence the total area is \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Option D misses one \(ab\) term. Exam tip: count the two \(ab\) rectangles separately in an area model.

The side lengths of the two squares are \(x+5\) and \(x+1\), so the difference of their areas is \((x+5)^2-(x+1)^2\). Applying the difference-of-squares identity gives \((x+5)^2-(x+1)^2=[(x+5)+(x+1)]\,[(x+5)-(x+1)]\). Thus, \((2x+6)\times4=8x+24\). Therefore, \(8x+24\) is correct. In \(4x+24\), the coefficient of the \(x\)-term is incorrectly calculated when multiplying \(2x+6\) by 4. Exam tip: for a difference of square areas, use \(A^2-B^2=(A+B)(A-B)\) directly.

The square model for side \\(x+3\\) contains a large square of area \\(x^2\\), one strip of area \\(3x\\), and a second strip of area \\(3x\\). If the small corner is not included, only these three regions remain. Their total area is \\(x^2+3x+3x=x^2+6x\\). The corner, which would have area \\(3^2=9\\), is deliberately excluded.

The complete square would be \\( (x+3)^2=x^2+6x+9\\), but the question asks for the area before the corner is added. Thus option D includes one extra region, while option A counts only one strip. Option C combines the large square with the corner but leaves out both strips. Therefore option B correctly describes the remaining area.

(10x) comes from two (5x) strips, so the corner must be (5^2=25). In exams, take half of the middle coefficient and square it.

(-6x) comes from two (-3x) strips, so the corner (3^2=9) is needed. In exams, square half the coefficient.

Each side of the square is \((a+1)\), so its total area is \((a+1)^2\). On expansion, \((a+1)^2=a^2+2a+1\). The four regions typically have areas \(a^2\), \(a\), \(a\), and \(1\), whose sum is \(a^2+2a+1\). Option \(a^2+a+1\) misses one region of area \(a\). Exam tip: add the areas of all smaller regions to check an area-model identity.

The tile model contains four parts of a complete square: an \\(x^2\\) tile, two rectangles each with area \\(3x\\), and a small square of area \\(9\\). Since \\(9=3^2\\), the side of that small square is \\(3\\). The large \\(x^2\\) tile has side \\(x\\), so placing the two strips around it gives a complete square with side \\(x+3\\).

The area check is \\(x^2+3x+3x+9=x^2+6x+9=(x+3)^2\\). Thus the side is not \\(x+9\\), because area 9 represents a side of 3, not 9. It is also not \\(x-3\\), since all the pieces are being assembled positively. Therefore option B is correct.

The area of the rectangle is the product of its side lengths: \((p+4)(p-4)\). Applying \((a+b)(a-b)=a^2-b^2\) gives \(p^2-4^2=p^2-16\). In \(p^2+16\), the sign of the constant term is incorrect; the middle terms cancel in a product of a sum and a difference. Exam tip: write \((x+a)(x-a)\) directly as \(x^2-a^2\).

In the area model of \((y+6)^2\), the large square is divided into one \(y\times y\) square, two \(y\times 6\) rectangles, and a small \(6\times 6\) corner square. Therefore, the area of the small corner is \(6\times 6=36\). \(6y\) is the area of only one rectangle, while \(12y\) is the total area of the two rectangles. Exam tip: in the area model of \((a+b)^2\), the small corner is always the square of the constant term, \(b^2\).

When the side \((m+2)\) is split into parts \(m\) and \(2\), two equal rectangles are formed. The area of each rectangle is \(m\times 2=2m\). Therefore, their total area is \(2m+2m=4m\). \(2m\) is the area of only one rectangle. Exam tip: in visual identity models, find each equal part first and then add them.

The area of the large square is \(b^2\), and the removed small square has area \(c^2\). Thus the remaining area is \(b^2-c^2\), which is called a difference of two squares. The identity for this expression is \(b^2-c^2=(b+c)(b-c)\). This means the same remaining area can be rearranged into a rectangle whose side lengths are \(b+c\) and \(b-c\).

Option A gives exactly these two side lengths. Multiplying them confirms the result: \((b+c)(b-c)=b^2-bc+bc-c^2=b^2-c^2\). The other choices either use the original parts separately or do not reproduce the required area. Therefore, the rectangle made by sides \(b+c\) and \(b-c\) is the correct model for the remaining region.

Since (4=2^2) and the middle term is (4x), the side is (x+2). Exam tip: check the square root of the last term.

In the rectangle model, the two mixed regions give the terms \(5r\) and \(3r\). Since they are like terms, \(5r+3r=8r\). It is not \(15r\), because multiplying 3 and 5 gives the constant term \(15\). Exam tip: combine only terms with the same variable and exponent.

((-5)^2=25), so the last term is positive. Exam tip: keep the square of a negative number positive.

The total area of the square model is \(u^2+10u+25\). Here, \(25=5^2\) and \(10u=2\times u\times5\). Therefore, \(u^2+10u+25=(u+5)^2\), so the side of the square is \(u+5\). In contrast, \((u-5)^2=u^2-10u+25\), which has a negative middle term. Exam tip: identify the middle term of a perfect-square trinomial as \(2ab\).