Question

7. The real solutions of 3^2 −+2=0 area. 1 and -2/3b. -1 and 2/3c. There are no real solutionsd. 1 is the only real solutione. -2/3 is the only real solution

Answer 1

Step 1:

In this question, we are given the quadratic equation:

`\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}`

Step 2:

Using the Quadratic Formulae, we have that:

`\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}`
`\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}`

Step 3:

From the calculations, the solutions have no real solutions.

We can further prove this using the Graphical method:

The graph of:

`3x^2\text{ - x + 2 = 0}`

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