Solve the equation: \$(3x^2 + 4) / (2x) = (x - 1) / 3\$.

The given equation

\$(3x^2 + 4) / (2x) = (x - 1) / 3\$

To solve this equation, we can start by cross-multiplying to eliminate the fractions then simplified

\$ 3(3x^2 + 4) = 2x(x - 1) \$

\$ 9x^2 - 2x^2 + 2x + 12 = 0 \$

\$ 7x^2 + 2x + 12 = 0 \$

Now we have a standard quadratic equation. We can solve it using the quadratic formula:

\$ x = (-b ± \sqrt(b^2 - 4ac)) / (2a) \$

In this case, a = 7, b = 2, and c = 12. Plugging these values into the quadratic formula, then we have:

\$ x = (-(2) ± \sqrt((2)^2 - 4(7)(12))) / (2(7)) \$

\$ x = (-2 ± \sqrt(-332)) / 14 \$

Since the discriminant is negative, there are no real solutions to this equation. The solutions are complex numbers:

\$ x = (-2 + 2i\sqrt83) / 14 and x = (-2 - 2i\sqrt83) / 14 \$

\$ x = (-1 + i\sqrt83) / 7 and x = (-1 - i\sqrt83) / 7 \$

Therefore, the solutions to the equation \$(3x^2 + 4) / (2x) = (x - 1) / 3\$ are \$ x = (-1 + i\sqrt83) / 7 and x = (-1 - i\sqrt83) / 7 \$.

This option presents the correct solutions we obtained. It shows two possible values of x

\$ (-1 + i\sqrt83) / 7 and (-1 - i\sqrt83) / 7 \$

This option is incorrect. The square root of a negative number results in an imaginary unit, not a real number

\$ ((2) + \sqrt(-332))/14\$ and \$ ((2) - \sqrt(-332))/14\$

Similar to option (B), this choice is incorrect for the same reason. The square root of -332 is imaginary, and the expression wouldn’t represent the solution form we obtained

\$ ((-2) + \sqrt(-332))/7\$ and \$ ((-2) - \sqrt(-332))/7\$

We arrived at a solution with a real part and an imaginary part, which aligns with option (A)

None of these above

Option

A

Can you summarize what you’ve understood in the above steps?