SOLUTION:
Step 1:
In this question, we are given the quadratic equation:
[tex]\begin{gathered} 3x^2\text{ -x + 2 = 0} \\ \text{comparing with:} \\ ax^2\text{ + bx + c = 0, we have that:} \\ a\text{ = 3, b = -1 , c = 2} \end{gathered}[/tex]Step 2:
Using the Quadratic Formulae, we have that:
[tex]\begin{gathered} x\text{ = }\frac{-b\text{ }\pm\text{ }\sqrt[]{b^2-\text{ 4ac}}}{2a} \\ u\sin g\text{ } \\ a\text{ = 3, b -1 , c = 2 } \\ we\text{ have that:} \end{gathered}[/tex][tex]\begin{gathered} x=\text{ }\frac{-(\text{ -1 ) }\pm\text{ }\sqrt[]{(-1)^2\text{ - 4(3)(2)}}}{2(3)} \\ \text{x = }\frac{1\text{ }\pm\sqrt[]{1-24}}{6} \\ \text{x =}\frac{1\text{ }\pm\sqrt[]{-23}}{6} \\ x\text{ =}\frac{1+\sqrt[]{23}\text{ i}}{2}\text{ or x = }\frac{1-\sqrt[]{23\text{ }}\text{ i}}{2} \end{gathered}[/tex]Step 3:
From the calculations, the solutions have no real solutions.
We can further prove this using the Graphical method:
The graph of:
[tex]3x^2\text{ - x + 2 = 0}[/tex]is as shown below:
CONCLUSION:
From the calculations and from the graph above, we can see clearly that:
The final answer is:
There are no real solutions --- OPTION C
