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