Use the quadratic formula to solve the eq \[ 4 x^{2}-4 x=-7 \]

To solve the equation \(4x^2 - 4x + 7 = 0\), we'll first rewrite it properly. Here, you must bring the \(-7\) to the left side, giving us \(4x^2 - 4x + 7 = 0\). Now, we'll identify the coefficients: \(a = 4\), \(b = -4\), and \(c = 7\). The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Plugging in the values: \[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 4 \cdot 7}}{2 \cdot 4} \] This simplifies to: \[ x = \frac{4 \pm \sqrt{16 - 112}}{8} \] Calculating the discriminant: \[ 16 - 112 = -96 \] Since the discriminant is negative, we will have complex solutions. Continuing with that: \[ x = \frac{4 \pm \sqrt{-96}}{8} = \frac{4 \pm 4i\sqrt{6}}{8} = \frac{1 \pm i\sqrt{6}}{2} \] So the solutions to the equation are: \[ x = \frac{1}{2} + \frac{i\sqrt{6}}{2} \quad \text{and} \quad x = \frac{1}{2} - \frac{i\sqrt{6}}{2} \]