Chapter 8.7, Problem 57PFA

To determine

To calculate the length of each side of square.

Answer to Problem 57PFA

  $4x+5$

Explanation of Solution

Given:

  $\text{Area of square = }16x^2+40x+25$

Formula used:

All the sides of a square is equal.

  $\text{Area of square = }{a}^2$

Roots of the polynomial of the form $a{x}^2+bx+c=a(x-x_1)(x-x_2)$ are

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

Consider the given expression,

  $\Rightarrow 16x^2+40x+25$

Substituting the formula for $a=16,b=40,c=25$

  $\begin{array}{l}\Rightarrow x=\frac{-40\pm \sqrt{{40}^2-4(16)(25)}}{2(16)}\\ \Rightarrow x=\frac{-40\pm \sqrt{1600-1600}}{32}\\ \Rightarrow x=\frac{-40}{32}=\frac{-5}{4}\end{array}$

Therefore, the roots are

  $\Rightarrow x_1=x_2=\frac{-5}{4}$

Thus, the polynomial can be factored as

  $\begin{array}{l}\Rightarrow 16x^2+40x+25=(x-(\frac{-5}{4}))(x-(\frac{-5}{4}))\\ \Rightarrow 16x^2+40x+25=(x+\frac{5}{4})(x+\frac{5}{4})\\ \Rightarrow 16x^2+40x+25=(4x+5)(4x+5)\\ $ \Rightarrow 16x^2+40x+25={(4x+5)}^2$\end{array}$

The area of square is $x^2$

Equating the area of square and the factored expression,

  $\begin{array}{l}$ \Rightarrow x^2={(4x+5)}^2$\Rightarrow x=(4x+5)\end{array}$

Since all the sides of a square are equal length of each side is $4x+5.$

Conclusion:

Therefore, the side of the square is $4x+5.$