12x^2-4x-1=0

To solve the quadratic equation \(12x^2 - 4x - 1 = 0\), we can use the quadratic formula:

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

where \(a = 12\), \(b = -4\), and \(c = -1\).

First, we need to calculate the discriminant (\(b^2 - 4ac\)):

\[ b^2 - 4ac = (-4)^2 - 4 \cdot 12 \cdot (-1) = 16 + 48 = 64 \]

Now we can substitute the values into the quadratic formula:

\[ x = \frac{-(-4) \pm \sqrt{64}}{2 \cdot 12} \] \[ x = \frac{4 \pm 8}{24} \]

This gives us two possible solutions:

1. \(x = \frac{4 + 8}{24} = \frac{12}{24} = \frac{1}{2}\)
2. \(x = \frac{4 - 8}{24} = \frac{-4}{24} = -\frac{1}{6}\)

Thus, the solutions to the equation \(12x^2 - 4x - 1 = 0\) are:

\[ x = \frac{1}{2} \quad \text{and} \quad x = -\frac{1}{6} \]