Questions: Use the quadratic formula to find the complex solutions of the quadratic equation. Give your answer(s) in a bi form. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) x=

Solution Steps

To find the complex solutions of a quadratic equation using the quadratic formula, follow these steps:

1. Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation in the form \(ax^2 + bx + c = 0\).
2. Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
3. Calculate the discriminant \(b^2 - 4ac\). If it is negative, the solutions will be complex.
4. Compute the solutions using the formula, ensuring to handle the square root of a negative number to find the imaginary part.

The quadratic equation is given in the form \( ax^2 + bx + c = 0 \). For our case, we have:

• \( a = 1 \)
• \( b = 2 \)
• \( c = 3 \)

The discriminant \( D \) is calculated using the formula: \[ D = b^2 - 4ac \] Substituting the values: \[ D = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8 \]

Since the discriminant \( D = -8 \) is negative, the quadratic equation has complex solutions.

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] Substituting the values: \[ x = \frac{-2 \pm \sqrt{-8}}{2 \cdot 1} \] This simplifies to: \[ x = \frac{-2 \pm 2i\sqrt{2}}{2} \] Thus, we can further simplify: \[ x = -1 \pm i\sqrt{2} \]

Final Answer