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