1. If \( \alpha \) is a root of the equation \( 2 x(2 x+1)=1 \), find the other root.

To find the roots of the equation \( 2 x(2 x + 1) = 1 \), we first need to rearrange the equation into standard polynomial form. 1. Start with the given equation: \[ 2 x(2 x + 1) = 1 \] 2. Expand the left side: \[ 4 x^2 + 2 x = 1 \] 3. Rearrange the equation to set it to zero: \[ 4 x^2 + 2 x - 1 = 0 \] Now we have a quadratic equation in the standard form \( ax^2 + bx + c = 0 \) where \( a = 4 \), \( b = 2 \), and \( c = -1 \). Next, we can use the quadratic formula to find the roots: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Let's calculate the discriminant \( b^2 - 4ac \): \[ b^2 - 4ac = 2^2 - 4 \cdot 4 \cdot (-1) = 4 + 16 = 20 \] Now we can find the roots using the quadratic formula: \[ x = \frac{-2 \pm \sqrt{20}}{2 \cdot 4} \] Calculating \( \sqrt{20} \): \[ \sqrt{20} = 2\sqrt{5} \] Substituting this back into the formula: \[ x = \frac{-2 \pm 2\sqrt{5}}{8} = \frac{-1 \pm \sqrt{5}}{4} \] Thus, the two roots are: \[ x_1 = \frac{-1 + \sqrt{5}}{4}, \quad x_2 = \frac{-1 - \sqrt{5}}{4} \] If \( \alpha \) is one of the roots, the other root is: \[ \text{Other root} = \frac{-1 - \sqrt{5}}{4} \] Therefore, the other root is: \[ \frac{-1 - \sqrt{5}}{4} \]