Solve the quadratic equation by completing the square. \[ x^{2}+18 x+71=0 \] First, choose the appropriate form and fill in the blanks with the correct nur Then, solve the equation. Simplify your answer as much as possible. If there is more than one solution, separate them with commas. Form: \( \quad(x+\square)^{2}=\square \) Solution: \( x=\square \)

To solve the quadratic equation \( x^{2} + 18x + 71 = 0 \) by completing the square, follow these steps: ### Step 1: Arrange the Equation Start by moving the constant term to the other side of the equation: \[ x^{2} + 18x = -71 \] ### Step 2: Complete the Square To complete the square, take half of the coefficient of \( x \), which is \( \frac{18}{2} = 9 \), and then square it, yielding \( 9^2 = 81 \). Add this value to both sides of the equation: \[ x^{2} + 18x + 81 = -71 + 81 \] \[ (x + 9)^2 = 10 \] ### Filled Form \[ (x + \boxed{9})^{2} = \boxed{10} \] ### Step 3: Solve for \( x \) Take the square root of both sides: \[ x + 9 = \pm \sqrt{10} \] \[ x = -9 \pm \sqrt{10} \] ### Solutions \[ x = -9 + \sqrt{10}, \quad x = -9 - \sqrt{10} \] ### Final Answer Form: \[ (x + \boxed{9})^{2} = \boxed{10} \] Solution: \[ x = -9 + \sqrt{10},\ -9 - \sqrt{10} \]