Chapter 8.7, Problem 42PPE

To determine

To find: The expression in representing the difference of the areas of the windows.

Answer to Problem 42PPE

The value is $A=\left(6x-5\right)\left(4x+13\right)$

Explanation of Solution

Formula used:

Substitution method is used.

Calculation:

Consider the area of the two square windows is as follows,

  $\begin{array}{l}{A}_1=25x^2+40x+16\\ A_2=x^2-18x+81\end{array}$

The above equation can be rewritten as follows,

  $\begin{array}{l}{A}_1=25x^2+40x+16\\ A_1={\left(5x+4\right)}^2\\ A_2=x^2-18x+81\\ A_2={\left(x-9\right)}^2\end{array}$

The difference between the areas of the two square windows is calculated as follows:

  $\begin{array}{l}A=\left(25x^2+40x+16\right)-\left(x^2-18x+81\right)\\ =25x^2+40x+16-x^2+18x-81\\ A=24x^2+58x-65\end{array}$

The Second method to calculate the difference between the areas of the two square windows is as follows,

  $\begin{array}{l}A={\left(5x+4\right)}^2-{\left(x-9\right)}^2\\ =\left(5x+4+x-9\right)\left(5x+4-x+9\right)\\ A=\left(6x-5\right)\left(4x+13\right)\end{array}$

Without using factor difference between the areas of the two square windows is calculated by the above first method, so the area will be equal to $24x^2+58x-65$

With use of factored form of areas, the difference between the areas of the two square windows is calculated as follows,

  $\begin{array}{l}A={\left(5x+4\right)}^2-{\left(x-9\right)}^2\\ =\left(5x+4+x-9\right)\left(5x+4-x+9\right)\\ A=\left(6x-5\right)\left(4x+13\right)\end{array}$

Conclusion:

The value is $A=\left(6x-5\right)\left(4x+13\right)$