Chapter 2.7, Problem 82E

To determine

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

Answer to Problem 82E

  (0.219, 0), (2.281, 0)

Explanation of Solution

Given:

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

Calculation for graph:

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

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 $x=0.219,\mathrm{2.281.}$

Hence, the x intercept is (0.219, 0) and (2.281, 0).

Verification:

  $\Rightarrow y=x-1-\frac{2}{2x-3}$

To find x intercepts, put y = 0,

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

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,

  $\begin{array}{l}\Rightarrow x=\frac{-(-5)\pm \sqrt{{(-5)}^2-4(2)(1)}}{2(2)}\\ \Rightarrow x=\frac{5\pm \sqrt{25-8}}{4}\\ \Rightarrow x=\frac{5\pm \sqrt{17}}{4}\\ \Rightarrow x=\frac{5+\sqrt{17}}{4},\frac{5-\sqrt{17}}{4}\\ \Rightarrow x=2.281,0.219\end{array}$

So, the x intercept is (0.219, 0) and (2.281, 0).

Conclusion:

Therefore, the x intercept of given equation is (0.219, 0) and (2.281, 0).