Chapter 1.3, Problem 60E

Solution

To find: the measure of length and width of rectangle.

Length of rectangle is 7 units and width is 12 units.

Given:

Area of rectangle is 84 square units.

Concept Used:

Factoring method to find the roots of the quadratic equation.

Calculation:

From the figure:

Length of rectangle is $(x+2)\text{ unit}$ .

And width of rectangle is $(x+7)\text{ unit}$ .

Formula of Area of rectangle $=\text{length }\times \text{ width}$ .

  $\therefore \left(x+2\right)\left(x+7\right)=84$

Solving the above equation:

  $\begin{array}{c}x\left(x+7\right)+2\left(x+7\right)=84\\ x^2+7x+2x+14=84\\ x^2+9x+14=84\\ x^2+9x+14-84=0\\ x^2+9x-70=0\end{array}$

Applying the factoring method of solving quadratic equation:

  $\begin{array}{c}{x}^2+9x-70=0\\ x^2+14x-5x-70=0\\ x\left(x+14\right)-5\left(x+14\right)=0\\ \left(x+14\right)\left(x-5\right)=0\end{array}$

Here, using Zero product property:

  $\begin{array}{c}x+14=0\text{or}x-5=0\\ x=-14\text{or}x=5\end{array}$

Here, Neglecting the value of $x=-14$ , As distance cannot be negative.

  $\begin{array}{l}\therefore \text{ length = 5+2}\\ =7\text{ units}\end{array}$

Also,

  $\begin{array}{l}\text{width = 5+2}\\ =12\text{ units}\end{array}$

Therefore, length of rectangle is 7 units and width is 12 units.

Conclusion:

The roots of the quadratic equation are $-6,-1$ .