Let's evaluate the expression \(\frac{2x^2 + 4x - 6}{x - 1}\) for different values of \(x\), filling in the missing values in the table step-by-step.
### Step-by-Step Calculation:
1. When \(x = 0\):
[tex]\[
\frac{2(0)^2 + 4(0) - 6}{0 - 1} = \frac{-6}{-1} = 6
\][/tex]
So, the value of the expression is \(6\).
2. When \(x = 0.5\):
[tex]\[
\frac{2(0.5)^2 + 4(0.5) - 6}{0.5 - 1} = \frac{2(0.25) + 2 - 6}{-0.5} = \frac{0.5 + 2 - 6}{-0.5} = \frac{-3.5}{-0.5} = 7
\][/tex]
So, the value of the expression is \(7\).
3. When \(x = 0.9\):
[tex]\[
\frac{2(0.9)^2 + 4(0.9) - 6}{0.9 - 1} = \frac{2(0.81) + 3.6 - 6}{-0.1} = \frac{1.62 + 3.6 - 6}{-0.1} = \frac{-0.78}{-0.1} = 7.8
\][/tex]
So, the value of the expression is approximately \(7.8\).
4. When \(x = 0.99\):
[tex]\[
\frac{2(0.99)^2 + 4(0.99) - 6}{0.99 - 1} = \frac{2(0.9801) + 3.96 - 6}{-0.01} = \frac{1.9602 + 3.96 - 6}{-0.01} = \frac{-0.0798}{-0.01} = 7.98
\][/tex]
So, the value of the expression is approximately \(7.98\).
5. When \(x = 0.999\):
[tex]\[
\frac{2(0.999)^2 + 4(0.999) - 6}{0.999 - 1} = \frac{2(0.998001) + 3.996 - 6}{-0.001} = \frac{1.996002 + 3.996 - 6}{-0.001} = \frac{-0.007998}{-0.001} = 7.998
\][/tex]
So, the value of the expression is approximately \(7.998\).
6. When \(x = 1.001\):
[tex]\[
\frac{2(1.001)^2 + 4(1.001) - 6}{1.001 - 1} = \frac{2(1.002001) + 4.004 - 6}{0.001} = \frac{2.004002 + 4.004 - 6}{0.001} = \frac{0.008002}{0.001} = 8.002
\][/tex]
So, the value of the expression is approximately \(8.002\).
7. When \(x = 1.01\):
[tex]\[
\frac{2(1.01)^2 + 4(1.01) - 6}{1.01 - 1} = \frac{2(1.0201) + 4.04 - 6}{0.01} = \frac{2.0402 + 4.04 - 6}{0.01} = \frac{0.0802}{0.01} = 8.02
\][/tex]
So, the value of the expression is approximately \(8.02\).
8. When \(x = 1.1\):
[tex]\[
\frac{2(1.1)^2 + 4(1.1) - 6}{1.1 - 1} = \frac{2(1.21) + 4.4 - 6}{0.1} = \frac{2.42 + 4.4 - 6}{0.1} = \frac{0.82}{0.1} = 8.2
\][/tex]
So, the value of the expression is approximately \(8.2\).
9. When \(x = 1.5\):
[tex]\[
\frac{2(1.5)^2 + 4(1.5) - 6}{1.5 - 1} = \frac{2(2.25) + 6 - 6}{0.5} = \frac{4.5 + 6 - 6}{0.5} = \frac{4.5}{0.5} = 9
\][/tex]
So, the value of the expression is \(9\).
10. When \(x = 2\):
[tex]\[
\frac{2(2)^2 + 4(2) - 6}{2 - 1} = \frac{2(4) + 8 - 6}{1} = \frac{8 + 8 - 6}{1} = \frac{10}{1} = 10
\][/tex]
So, the value of the expression is \(10\).
By compiling these results, we get the following table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$\frac{2 x^2 + 4 x - 6}{x - 1}$[/tex] \\
\hline
0 & 6 \\
\hline
0.5 & 7 \\
\hline
0.9 & 7.8 \\
\hline
0.99 & 7.98 \\
\hline
0.999 & 7.998 \\
\hline
1.001 & 8.002 \\
\hline
1.01 & 8.02 \\
\hline
1.1 & 8.2 \\
\hline
1.5 & 9 \\
\hline
2 & 10 \\
\hline
\end{tabular}
\][/tex]
