Solve the quadratic equation using the quadratic formula. Find only real solutions. If there are no real solutions, enter "DNE" in both boxes. \( 1+x-x^{2}=0 \)

To solve the quadratic equation \( 1 + x - x^{2} = 0 \) using the quadratic formula, follow these steps: 1. **Rewrite the Equation in Standard Form:** \[ -x^{2} + x + 1 = 0 \quad \text{or} \quad x^{2} - x - 1 = 0 \] Here, \( a = 1 \), \( b = -1 \), and \( c = -1 \). 2. **Apply the Quadratic Formula:** \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] Substitute the values of \( a \), \( b \), and \( c \): \[ x = \frac{-(-1) \pm \sqrt{(-1)^{2} - 4(1)(-1)}}{2(1)} = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2} \] 3. **Determine the Real Solutions:** \[ x = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{5}}{2} \] Both solutions are real numbers since the discriminant (\( \sqrt{5} \)) is positive. **Final Answer:** \[ x = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{5}}{2} \]