Step-by-step explanation:
To solve the quadratic equation \( x^2 + 9x + 18 = 0 \) by completing the square, follow these steps:
1. **Write down the equation:** \( x^2 + 9x + 18 = 0 \).
2. **Isolate the constant term:** Move the constant term (18) to the right side of the equation:
\[ x^2 + 9x = -18 \].
3. **Prepare to complete the square:** Take half the coefficient of \( x \) (which is 9), square it, and add/subtract it inside the equation. Half of 9 is \( \frac{9}{2} = 4.5 \), and squaring it gives \( \left(\frac{9}{2}\right)^2 = \frac{81}{4} \).
4. **Complete the square:** Add and subtract \( \frac{81}{4} \) inside the equation:
\[ x^2 + 9x + \frac{81}{4} = -18 + \frac{81}{4} \].
5. **Simplify the right-hand side:** Calculate \( -18 + \frac{81}{4} \):
\[ -18 = -\frac{72}{4} \], so \( -18 + \frac{81}{4} = -\frac{72}{4} + \frac{81}{4} = \frac{9}{4} \).
6. **Write it as a perfect square:** The left-hand side can be written as a perfect square:
\[ \left( x + \frac{9}{2} \right)^2 = \frac{9}{4} \].
7. **Solve for \( x \):** Take the square root of both sides:
\[ x + \frac{9}{2} = \pm \frac{3}{2} \].
8. **Solve for \( x \):** Solve for \( x \) by subtracting \( \frac{9}{2} \) from both sides:
\[ x = -\frac{9}{2} \pm \frac{3}{2} \].
9. **Find the solutions:** Simplify the solutions:
\[ x = -\frac{9}{2} + \frac{3}{2} = -3, \]
\[ x = -\frac{9}{2} - \frac{3}{2} = -6. \]
Therefore, the solutions to the quadratic equation \( x^2 + 9x + 18 = 0 \) are \( \boxed{-3 \text{ and } -6} \).
