X^2+3x+1=0

To solve the quadratic equation \( x^2 + 3x + 1 = 0 \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In this equation, the coefficients are: - \( a = 1 \) - \( b = 3 \) - \( c = 1 \) Now, we can substitute the values into the formula: 1. Calculate the discriminant (\( b^2 - 4ac \)): \[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 1 = 9 - 4 = 5 \] 2. Now substitute the values into the quadratic formula: \[ x = \frac{-3 \pm \sqrt{5}}{2 \cdot 1} \] 3. Simplifying that gives: \[ x = \frac{-3 \pm \sqrt{5}}{2} \] Thus, the solutions to the equation \( x^2 + 3x + 1 = 0 \) are: \[ x = \frac{-3 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{5}}{2} \]