Chapter 1.1, Problem 52E

Solution

a)

To find: The area of the given figure.

The required area is $x^2+bx$ .

Given Information:

The square shown in figure 1.

  

Figure 1

Concept Used:

The area of square is $\text{side}\times \text{side}$ and the area of rectangle is $l\times w$ .

Calculation:

Consider the 3 rectangles as shown in the figure 2.

  

Figure 2

The area of square 1 is $A_{S_1}=x^2$ , area of square 2 is $A_{S_2}=\frac{b}{2}\cdot x$ , area of square 3 is $A_{S_3}=\frac{b}{2}\cdot x$ .

Thus, the area of given figure is,

  $\begin{array}{c}{A}_S=A_{S_1}+A_{S_2}+A_{S_3}\\ =x^2+\frac{bx}{2}+\frac{bx}{2}\\ =x^2+bx\end{array}$

Therefore, the required area is $A_S=x^2+bx$ .

b)

To find: What area must be added to complete the large square

The the area that must be added is $\frac{b^2}{4}$ .

Given Information:

The square shown in figure 1.

  

Concept Used:

The area of square is $\text{side}\times \text{side}$ and the area of rectangle is $l\times w$ .

Calculation:

Consider the 3 rectangles as shown in the figure 2.

In order to complete the square, a square of side $\frac{b}{2}$ must be added.

The area of square of side $\frac{b}{2}$ is ${\left(\frac{b}{2}\right)}^2=\frac{b^2}{4}$ .

Therefore, the area that must be added is $\frac{b^2}{4}$ .

c)

To explain: How the algebraic formula for completing the square models the completing of the square in part (b).

Given Information:

The square shown in figure 1.

Explanation:

The area that must be added to complete the square is $\frac{b^2}{4}$ .

Now, let the area of the given figure be $x^2+bx=k$ .

Use the completing the squats method as follows.

Add square of half times the coefficient of $x$ , that is, ${\left(\frac{b}{2}\right)}^2=\frac{b^2}{4}$ to both sides of the equation $x^2+bx=k$ .

  $\begin{array}{c}{x}^2+bx+\frac{b^2}{4}=k+\frac{b^2}{4}\\ {\left(x+\frac{b}{2}\right)}^2=k+\frac{b^2}{4}\end{array}$

Thus, it follows that, in order to complete the square, $\frac{b^2}{4}$ must be added which is exactly the area of the missing square on the bottom right-hand side of the given figure.