Chapter 3.2, Problem 48HP

To determine

To explain which is it better to solve an equation; using algebraic methods or graphical method.

Explanation of Solution

Let the equation be: $x^2+x-6=0$

Now for the given equation, if it has been asked to find the values of x, then solving it algebraically would be more convenient.

Now,

  $\begin{array}{l}{x}^2+x-6=0\\ \Rightarrow x^2+3x-2x-6=0\\ \Rightarrow (x+3)(x-2)=0\\ \Rightarrow x=-3,2\end{array}$

Thus just by solving algebraically, the values of x could be easily obtained.

But, if the same equation would have been as such: $x^2+x-6=\left|x\right|$ and it was asked to find the number of solutions instead of actually finding the values of x. In this case graphical approach would have been much useful.

For this, assume two equations: $y=x^2+x-6$ and $y=\left|x\right|$

Calculation of graph:

Consider, $y=x^2+x-6$

Values of x	Values of y
0	-6
1	-4
2	0
-3	0
-1	-6

Using the above table the graph can be plotted.

Graph:

  

Calculation of graph:

Consider, $y=\left|x\right|$

Values of x	Values of y
0	0
1	0
2	2
-3	3
-1	1

Using the above table the graph can be plotted.

Graph:

  

Now to find the total number of solutions both the graph has to be plotted on the same plane and the number of point of intersection between the graphs has to be counted because the intersection of the graphs are the required solutions.

Plotting both the graphs on the same plane:

  Interpretation:

From the graph, it is clear that, there are two points of intersection, which means that there are two solutions for the given equation.

Conclusion:

Hence, the method should be selected considering the above analysis.