Understanding Complex Solutions

Oct 13, 2025 by pynkocean.com 109 views

Introduction

Quadratic equations are a fundamental concept in mathematics, and they can be solved using various methods, including factoring, the quadratic formula, and completing the square. However, when the solutions to a quadratic equation are complex numbers, it can be challenging to determine which equation has those solutions. In this article, we will explore how to identify the equation with complex solutions and provide a step-by-step guide on how to solve quadratic equations with complex roots.

Understanding Complex Solutions

Complex solutions, also known as complex roots, are solutions to a quadratic equation that involve imaginary numbers. These solutions are typically denoted by the expression $x = a \pm bi$ , where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$ . Complex solutions can be found using the quadratic formula, which is given by:

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

Analyzing the Given Solutions

The given solutions are $x = \frac{-3 \pm \sqrt{3} i}{2}$ . To determine which equation has these solutions, we need to analyze the coefficients of the quadratic equation. The solutions are in the form of $x = a \pm bi$ , where $a = -3/2$ and $b = \sqrt{3}/2$ . We can rewrite the solutions as:

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

Comparing the Solutions with the Options

Now, let's compare the given solutions with the options provided:

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

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

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

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

Solving Option A

To determine if option A is the correct equation, we can start by dividing the equation by 2 to simplify it:

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

Now, we can compare the coefficients of the simplified equation with the given solutions. The coefficient of $x^2$ is 1, the coefficient of $x$ is 3, and the constant term is $\frac{9}{2}$ . We can rewrite the solutions as:

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

Comparing the coefficients, we can see that the coefficient of $x$ is 3, which matches the given solutions. However, the constant term is $\frac{9}{2}$ , which does not match the given solutions.

Solving Option B

To determine if option B is the correct equation, we can start by dividing the equation by 1 to simplify it:

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

Now, we can compare the coefficients of the simplified equation with the given solutions. The coefficient of $x^2$ is 1, the coefficient of $x$ is 3, and the constant term is 12. We can rewrite the solutions as:

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

Comparing the coefficients, we can see that the coefficient of $x$ is 3, which matches the given solutions. However, the constant term is 12, which does not match the given solutions.

Solving Option C

To determine if option C is the correct equation, we can start by dividing the equation by 1 to simplify it:

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

Now, we can compare the coefficients of the simplified equation with the given solutions. The coefficient of $x^2$ is 1, the coefficient of $x$ is 3, and the constant term is 3. We can rewrite the solutions as:

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

Comparing the coefficients, we can see that the coefficient of $x$ is 3, which matches the given solutions. The constant term is 3, which matches the given solutions.

Solving Option D

To determine if option D is the correct equation, we can start by dividing the equation by 2 to simplify it:

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

Now, we can compare the coefficients of the simplified equation with the given solutions. The coefficient of $x^2$ is 1, the coefficient of $x$ is $\frac{3}{2}$ , and the constant term is $\frac{3}{2}$ . We can rewrite the solutions as:

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

Comparing the coefficients, we can see that the coefficient of $x$ is $\frac{3}{2}$ , which does not match the given solutions.

Conclusion

Based on the analysis, we can conclude that the correct equation is option C: $x^2 + 3x + 3 = 0$ . This equation has the solutions $x = \frac{-3 \pm \sqrt{3} i}{2}$ .

Final Answer

The final answer is option C: $x^2 + 3x + 3 = 0$ .

Introduction

Quadratic equations with complex solutions can be challenging to solve, but with the right approach, you can master this topic. In this article, we will provide a Q&A guide to help you understand and solve quadratic equations with complex solutions.

Q: What is a quadratic equation with complex solutions?

A: A quadratic equation with complex solutions is an equation of the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are real numbers, and the solutions involve imaginary numbers.

Q: How do I identify a quadratic equation with complex solutions?

A: To identify a quadratic equation with complex solutions, look for the following characteristics:

The equation has a negative discriminant ( $b^2 - 4ac < 0$ )
The solutions involve imaginary numbers ( $x = a \pm bi$ )
The equation has no real solutions

Q: What is the quadratic formula for complex solutions?

A: The quadratic formula for complex solutions is:

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

Q: How do I solve a quadratic equation with complex solutions?

A: To solve a quadratic equation with complex solutions, follow these steps:

Write the equation in the form $ax^2 + bx + c = 0$
Calculate the discriminant ( $b^2 - 4ac$ )
If the discriminant is negative, the equation has complex solutions
Use the quadratic formula to find the solutions
Simplify the solutions to the form $x = a \pm bi$

Q: What is the difference between real and complex solutions?

A: Real solutions are solutions that involve only real numbers, while complex solutions involve imaginary numbers. Complex solutions can be written in the form $x = a \pm bi$ , where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

Q: Can a quadratic equation have both real and complex solutions?

A: No, a quadratic equation can have either real or complex solutions, but not both.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, follow these steps:

Calculate the discriminant ( $b^2 - 4ac$ )
If the discriminant is positive, the equation has two real solutions
If the discriminant is zero, the equation has one real solution
If the discriminant is negative, the equation has two complex solutions

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions if the equation is inconsistent (e.g., $0 = 1$ ).

Q: How do I graph a quadratic equation with complex solutions?

A: To graph a quadratic equation with complex solutions, follow these steps:

Plot the real axis
Plot the imaginary axis
Plot the solutions in the form $x = a \pm bi$
Use a graphing calculator or software to visualize the graph

Conclusion

Quadratic equations with complex solutions can be challenging to solve, but with the right approach, you can master this topic. By following the steps outlined in this Q&A guide, you can identify and solve quadratic equations with complex solutions.

Final Tips

Practice solving quadratic equations with complex solutions to become proficient
Use a graphing calculator or software to visualize the graph
Check your work by plugging the solutions back into the original equation

Additional Resources

Online resources: Khan Academy, Mathway, Wolfram Alpha
Textbooks: "Algebra and Trigonometry" by Michael Sullivan, "College Algebra" by James Stewart
Online communities: Reddit's r/learnmath, Stack Exchange's Mathematics community