Questions: x^2 + 3x + 3 = 0

Solution Steps

To solve the quadratic equation \(x^2 + 3x + 3 = 0\), we can use the quadratic formula, which is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b = 3\), and \(c = 3\). We will calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots and then apply the formula to find the solutions.

For the quadratic equation \(x^2 + 3x + 3 = 0\), we identify the coefficients as follows:

• \(a = 1\)
• \(b = 3\)
• \(c = 3\)

We calculate the discriminant using the formula \(D = b^2 - 4ac\): \[ D = 3^2 - 4 \cdot 1 \cdot 3 = 9 - 12 = -3 \] Since the discriminant is negative (\(D < 0\)), the equation has two complex roots.

Using the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\), we find the roots: \[ x_1 = \frac{-3 + \sqrt{-3}}{2 \cdot 1} = \frac{-3 + \sqrt{3}i}{2} = -1.5 + 0.8660i \] \[ x_2 = \frac{-3 - \sqrt{-3}}{2 \cdot 1} = \frac{-3 - \sqrt{3}i}{2} = -1.5 - 0.8660i \]

Final Answer