Chapter 1.1, Problem 50E

Solution

To explain: Explain how the algebraic equation ${\left(x+b\right)}^2=x^2+2bx+b^2$ models the areas of the regions in the given geometric figure

Given Information:

The following geometrical figure.

  

Explanation:

Consider the whole square with side $x+b$ units.

Then the area of the square is ${\text{A}}_s\text{= }{\left(x+2\right)}^2$ ..... (1)

Now, consider the four squares as shown in the figure 1.

  

The area of square 1 is $A_{S_1}=x^2$ , area of square 2 is $A_{S_2}=b\cdot x$ , area of square 3 is $A_{S_3}=b\cdot x$ and the area of square 4 is $A_{S_4}=b^2$ .

Thus, the area of whole square is,

  $\begin{array}{c}{A}_S=A_{S_1}+A_{S_2}+A_{S_3}+A_{S_4}\\ =x^2+bx+bx+b^2\end{array}$

Simplify further.

  $A_S=x^2+2bx+b^2$ ..... (2)

Therefore, from equation (1) and (2) it follows that ${\left(x+b\right)}^2=x^2+2bx+b^2$ .