Find all solutions of the equation and express them in the form a + bi. (Enter your answers as a comma-separated list. Simplify your answer completely.) \[ 64 x^{2}+3=16 x \] \( x=\square \)

We start with the equation \[ 64x^2 + 3 = 16x. \] First, we bring all terms to one side: \[ 64x^2 - 16x + 3 = 0. \] This is a quadratic equation in the form \( ax^2 + bx + c = 0 \) with \( a = 64 \), \( b = -16 \), and \( c = 3 \). Using the quadratic formula, \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] we substitute the values: \[ x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(64)(3)}}{2(64)}. \] Simplify step by step: 1. Evaluate \(-(-16)\): \[ -(-16) = 16. \] 2. Compute the discriminant: \[ (-16)^2 - 4(64)(3) = 256 - 768 = -512. \] 3. Write the square root of the discriminant: \[ \sqrt{-512} = i\sqrt{512} = i\sqrt{256 \cdot 2} = 16i\sqrt{2}. \] Thus, the expression for \( x \) becomes: \[ x = \frac{16 \pm 16i\sqrt{2}}{128}. \] Simplify by factoring out \( 16 \): \[ x = \frac{16(1 \pm i\sqrt{2})}{128} = \frac{1 \pm i\sqrt{2}}{8}. \] So the solutions in the form \( a+bi \) are: \[ x = \frac{1}{8} + \frac{\sqrt{2}}{8}i, \quad x = \frac{1}{8} - \frac{\sqrt{2}}{8}i. \]