Chapter 1.3, Problem 61E

Solution

To find: The value of $x$ .

The correct value of $x\text{ is 3}\text{.}$

Given:

Area of triangle is 42 square units.

Concept Used:

Factoring method to find the roots of the quadratic equation.

Calculation:

From the figure:

Base of triangle is $(2x+8)\text{ unit}$ .

And height of triangle is $\left(x+3\right)\text{unit}$ .

Formula of Area of triangle $=\frac{1}{2}\text{base }\times \text{ height}$ .

  $\therefore \frac{1}{2}\left(2x+8\right)\left(x+3\right)=42$

Solving the above equation:

  $\begin{array}{c}\frac{1}{2}\left(2x+8\right)\left(x+3\right)=42\\ \left(2x+8\right)\left(x+3\right)=42\times 2\\ 2x\left(x+3\right)+8\left(x+3\right)=84\\ 2x^2+6x+8x+24=84\\ 2x^2+14x+24-84=0\\ 2x^2+14x-60=0\end{array}$

Applying the factoring method of solving quadratic equation:

  $\begin{array}{c}2x^2+14x-60=0\\ 2x^2+20x-6x-60=0\\ 2x\left(x+10\right)-6\left(x+10\right)=0\\ \left(x+10\right)\left(2x-6\right)=0\end{array}$

Here, using Zero product property:

  $\begin{array}{c}x+10=0\text{or}2x-6=0\\ x=-10\text{or}x=\frac{6}{2}\\ x=-10\text{or}x=3\end{array}$

Here, Neglecting the value of $x=-10$ , As distance cannot be negative.

Therefore, the correct value of $x\text{ is 3}\text{.}$

Conclusion:

The correct value of $x\text{ is 3}\text{.}$