Chapter 2.7, Problem 81E

To determine

To estimate the x intercept of the equation $y=x+3-\frac{2}{2x-1}$ using graph and verifying it numerically.

Answer to Problem 81E

  (-3.266, 0), (0.766, 0)

Explanation of Solution

Given:

Equation: $y=x+3-\frac{2}{2x-1}$

Calculation for graph:

Consider $y=x+3-\frac{2}{2x-1}$

Values of x	Values of y
0	5
1	2
-1	2.666
2	4.333
-2	1.4

By taking different values of x, the graph can be plotted.

Graph:

  

Interpretation:

From above graph, it is clear that the graph meets x-axis at $x=-3.266,\mathrm{0.766.}$

Hence, the x intercept is (-3.266, 0) and (0.766, 0).

Verification:

  $\Rightarrow y=x+3-\frac{2}{2x-1}$

To find x intercepts, put y = 0,

  $\begin{array}{l}\Rightarrow y=0\\ \Rightarrow x+3-\frac{2}{2x-1}=0\\ \Rightarrow x+3=\frac{2}{2x-1}\\ \Rightarrow (x+3)(2x-1)=2\\ \Rightarrow 2x^2-x+6x-3=2\\ \Rightarrow 2x^2-x+6x-3-2=0\\ \Rightarrow 2x^2+5x-5=0\end{array}$

The roots of a quadratic equation $a{x}^2+bx+c=0$ can be found using the formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Applying above formula,

  $\begin{array}{l}\Rightarrow x=\frac{-5\pm \sqrt{{(5)}^2-4(2)(-5)}}{2(2)}\\ \Rightarrow x=\frac{-5\pm \sqrt{25+40}}{4}\\ \Rightarrow x=\frac{-5\pm \sqrt{65}}{4}\\ \Rightarrow x=\frac{-5+\sqrt{65}}{4},\frac{-5-\sqrt{65}}{4}\\ \Rightarrow x=0.766,-3.266\end{array}$

So, the x intercept is (-3.266, 0) and (0.766, 0).

Conclusion:

Therefore, the x intercept of given equation is (-3.266, 0) and (0.766, 0).