Solve the equation using the quadratic formula. \( x^{2}+5 x+5=0 \)

To solve the quadratic equation \( x^{2} + 5x + 5 = 0 \) using the **quadratic formula**, follow these steps: ### Quadratic Formula The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^{2} + bx + c = 0 \). ### Identifying the Coefficients For the equation \( x^{2} + 5x + 5 = 0 \): - \( a = 1 \) - \( b = 5 \) - \( c = 5 \) ### Plugging into the Formula Substitute the coefficients into the quadratic formula: \[ x = \frac{-5 \pm \sqrt{5^{2} - 4 \cdot 1 \cdot 5}}{2 \cdot 1} \] ### Simplifying Inside the Square Root Calculate the discriminant (\( D \)): \[ D = b^{2} - 4ac = 25 - 20 = 5 \] ### Solving for \( x \) Now, substitute \( D = 5 \) back into the formula: \[ x = \frac{-5 \pm \sqrt{5}}{2} \] ### Final Solutions The equation has two solutions: \[ x = \frac{-5 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-5 - \sqrt{5}}{2} \] ### Approximate Decimal Values (Optional) If you need decimal approximations: - \( \sqrt{5} \approx 2.236 \) So, \[ x \approx \frac{-5 + 2.236}{2} = \frac{-2.764}{2} = -1.382 \] \[ x \approx \frac{-5 - 2.236}{2} = \frac{-7.236}{2} = -3.618 \] ### Conclusion The solutions to the equation \( x^{2} + 5x + 5 = 0 \) are: \[ x = \frac{-5 \pm \sqrt{5}}{2} \] Which approximately are: \[ x \approx -1.382 \quad \text{and} \quad x \approx -3.618 \]