Which Equation Has The Solutions $x=\frac{-3 \pm \sqrt{3 I}}{2}$?A. $2x^2 + 6x + 9 = 0$ B. $x^2 + 3x + 12 = 0$ C. $x^2 + 3x + 3 = 0$ D. $2x^2 + 6x + 3 = 0$

Introduction

Quadratic equations are a fundamental concept in mathematics, and they can be used to model a wide range of real-world phenomena. In this article, we will explore the solutions to a quadratic equation that involve complex numbers. We will examine the given solutions and determine which equation they satisfy.

Complex Solutions to Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $x$ is the variable. The solutions to a quadratic equation can be found using the quadratic formula:

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

When the discriminant $b^2 - 4ac$ is negative, the solutions to the quadratic equation will be complex numbers.

Given Solutions

The given solutions are:

$$x = \frac{-3 \pm \sqrt{3i}}{2}$$

These solutions involve complex numbers, specifically the imaginary unit $i$, which is defined as the square root of $-1$. The solutions can be rewritten as:

$$x = \frac{-3 \pm i\sqrt{3}}{2}$$

Analyzing the Solutions

To determine which equation the given solutions satisfy, we need to analyze the solutions and compare them to the equations provided in the options.

Option A: $2x^2 + 6x + 9 = 0$

Let's start by analyzing Option A. We can rewrite the equation as:

$$2x^2 + 6x + 9 = 0$$

$$\Rightarrow x^2 + 3x + \frac{9}{2} = 0$$

Using the quadratic formula, we get:

$$x = \frac{-3 \pm \sqrt{3}}{\sqrt{2}}$$

This solution does not match the given solutions.

Option B: $x^2 + 3x + 12 = 0$

Next, let's analyze Option B. We can rewrite the equation as:

$$x^2 + 3x + 12 = 0$$

Using the quadratic formula, we get:

$$x = \frac{-3 \pm \sqrt{33}}{2}$$

This solution does not match the given solutions.

Option C: $x^2 + 3x + 3 = 0$

Now, let's analyze Option C. We can rewrite the equation as:

$$x^2 + 3x + 3 = 0$$

Using the quadratic formula, we get:

$$x = \frac{-3 \pm \sqrt{3}}{2}$$

This solution matches the given solutions.

Option D: $2x^2 + 6x + 3 = 0$

Finally, let's analyze Option D. We can rewrite the equation as:

$$2x^2 + 6x + 3 = 0$$

$$\Rightarrow x^2 + 3x + \frac{3}{2} = 0$$

Using the quadratic formula, we get:

$$x = \frac{-3 \pm \sqrt{\frac{3}{2}}}{2}$$

This solution does not match the given solutions.

Conclusion

Based on our analysis, we can conclude that the equation that has the solutions $x = \frac{-3 \pm \sqrt{3i}}{2}$ is:

$x^2 + 3x + 3 = 0$

This equation satisfies the given solutions, and it is the correct answer.

Final Answer

The final answer is:

• C. $x^2 + 3x + 3 = 0$
  Quadratic Equations with Complex Solutions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the solutions to a quadratic equation that involve complex numbers. We analyzed the given solutions and determined which equation they satisfy. In this article, we will provide a Q&A guide to help you better understand quadratic equations with complex solutions.

Q: What are quadratic equations?

A: Quadratic equations are polynomial equations of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to find the solutions to a quadratic equation. It is given by:

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

Q: What happens when the discriminant is negative?

A: When the discriminant $b^2 - 4ac$ is negative, the solutions to the quadratic equation will be complex numbers.

Q: What are complex numbers?

A: Complex numbers are numbers that have both real and imaginary parts. The imaginary unit $i$ is defined as the square root of $-1$. Complex numbers can be written in the form:

$$a + bi$$

where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

Q: How do I determine which equation satisfies the given solutions?

A: To determine which equation the given solutions satisfy, you need to analyze the solutions and compare them to the equations provided in the options. You can use the quadratic formula to find the solutions to each equation and compare them to the given solutions.

Q: What are some common mistakes to avoid when working with complex solutions?

A: Some common mistakes to avoid when working with complex solutions include:

• Not checking the discriminant to see if it is negative
• Not using the correct formula to find the solutions
• Not simplifying the solutions correctly
• Not checking the solutions to see if they satisfy the original equation

Q: How do I simplify complex solutions?

A: To simplify complex solutions, you can use the following steps:

1. Multiply the numerator and denominator by the conjugate of the denominator
2. Simplify the expression
3. Write the solution in the form $a + bi$

Q: What are some real-world applications of quadratic equations with complex solutions?

A: Quadratic equations with complex solutions have many real-world applications, including:

• Electrical engineering: Quadratic equations with complex solutions are used to model electrical circuits and analyze their behavior.
• Signal processing: Quadratic equations with complex solutions are used to analyze and process signals in fields such as audio and image processing.
• Control systems: Quadratic equations with complex solutions are used to model and analyze control systems in fields such as robotics and aerospace engineering.

Conclusion

In this article, we provided a Q&A guide to help you better understand quadratic equations with complex solutions. We covered topics such as the quadratic formula, complex numbers, and real-world applications. We hope this guide has been helpful in your studies and will provide you with a better understanding of quadratic equations with complex solutions.

Final Answer

The final answer is:

• C. $x^2 + 3x + 3 = 0$