Questions: Use the quadratic formula to solve the equation x^2 + 6x + 18 = 0. Enter multiple answers as a list separated by commas. Example: 2 + 2i, 2 - 2i

Solution Steps

To solve the quadratic equation \(x^2 + 6x + 18 = 0\), we will use the quadratic formula, which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \(a\), \(b\), and \(c\) are the coefficients of the equation \(ax^2 + bx + c = 0\). In this case, \(a = 1\), \(b = 6\), and \(c = 18\). We will calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots and then compute the roots using the formula.

For the quadratic equation \(x^2 + 6x + 18 = 0\), we identify the coefficients as follows:

• \(a = 1\)
• \(b = 6\)
• \(c = 18\)

We calculate the discriminant using the formula \(D = b^2 - 4ac\): \[ D = 6^2 - 4 \cdot 1 \cdot 18 = 36 - 72 = -36 \] Since the discriminant is negative, this indicates that the roots are complex.

Using the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\), we find the roots: \[ x = \frac{-6 \pm \sqrt{-36}}{2 \cdot 1} = \frac{-6 \pm 6i}{2} = -3 \pm 3i \] Thus, the two roots are: \[ x_1 = -3 + 3i \quad \text{and} \quad x_2 = -3 - 3i \]

Final Answer