Question

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

Answer 1

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:

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