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:

\[ x = \frac{-3 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{5}}{2} \]