Solve -2x ( 6x-5)=4

To solve the equation \(-2x(6x - 5) = 4\), we can start by expanding the left side:

\[ -2x(6x - 5) = -2x \cdot 6x + (-2x)(-5) \] \[ = -12x^2 + 10x \]

Now, the equation becomes:

\[ -12x^2 + 10x = 4 \]

Next, we will move all terms to one side of the equation to set it to zero:

\[ -12x^2 + 10x - 4 = 0 \]

To make calculations easier, we can multiply the entire equation by \(-1\):

\[ 12x^2 - 10x + 4 = 0 \]

Now we will use the quadratic formula to solve for \(x\). The quadratic formula is:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our equation, \(a = 12\), \(b = -10\), and \(c = 4\). Now we will calculate the discriminant:

\[ b^2 - 4ac = (-10)^2 - 4 \cdot 12 \cdot 4 \] \[ = 100 - 192 \] \[ = -92 \]

Since the discriminant is negative, this means that there are no real solutions to the equation. The solutions will be complex.

Now we can continue and find the complex solutions:

\[ x = \frac{-(-10) \pm \sqrt{-92}}{2 \cdot 12} \] \[ = \frac{10 \pm \sqrt{92}i}{24} \]

We can simplify \(\sqrt{92}\) as follows:

\[ \sqrt{92} = \sqrt{4 \cdot 23} = 2\sqrt{23} \]

Thus, the solutions for \(x\) are:

\[ x = \frac{10 \pm 2\sqrt{23}i}{24} \]

Simplifying further gives:

\[ x = \frac{5 \pm \sqrt{23}i}{12} \]

So the final solutions are:

\[ x = \frac{5}{12} + \frac{\sqrt{23}}{12}i \quad \text{and} \quad x = \frac{5}{12} - \frac{\sqrt{23}}{12}i \]