Select The Correct Answer.One Solution To A Quadratic Function, H H H , Is Given: − 4 + 7 I -4+7i − 4 + 7 I .Which Statement Is True?A. Function H H H Has No Other Solutions.B. The Other Solution To Function H H H Is − 4 − 7 I -4-7i − 4 − 7 I .C. The

Understanding Complex Roots

When dealing with quadratic equations, we often encounter complex roots. These roots are essential in understanding the behavior of quadratic functions and their applications in various fields. In this article, we will explore the concept of complex solutions of quadratic equations and determine the correct statement regarding a given solution.

Complex Numbers

Before diving into the world of complex roots, it's essential to understand complex numbers. A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$. Complex numbers can be represented graphically on a complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

Quadratic Equations with Complex Roots

A quadratic equation is a polynomial equation of degree two, which can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers, and $a \neq 0$. When solving quadratic equations, we often encounter complex roots, which are solutions that involve imaginary numbers.

The Given Solution

The given solution to the quadratic function $h$ is $-4 + 7i$. This solution is a complex number, which means it has both a real part and an imaginary part.

Analyzing the Options

Now, let's analyze the given options to determine the correct statement.

Option A: Function $h$ has no other solutions.

This statement is incorrect because a quadratic function with complex roots always has two solutions. The given solution $-4 + 7i$ is one of the solutions, and the other solution must be its conjugate.

Option B: The other solution to function $h$ is $-4 - 7i$.

This statement is correct because the other solution to the quadratic function $h$ is indeed the conjugate of the given solution, which is $-4 - 7i$. The conjugate of a complex number is obtained by changing the sign of the imaginary part.

Option C: The solutions to function $h$ are $-4 + 7i$ and $-4 - 7i$.

This statement is incorrect because the solutions to the quadratic function $h$ are not $-4 + 7i$ and $-4 - 7i$. The correct solutions are $-4 + 7i$ and its conjugate, which is $-4 - 7i$.

Conclusion

In conclusion, the correct statement is that the other solution to the quadratic function $h$ is $-4 - 7i$. This is because the solutions to a quadratic equation with complex roots always come in conjugate pairs.

Why Conjugate Pairs?

Conjugate pairs are essential in understanding complex roots because they ensure that the quadratic function has real coefficients. When a quadratic function has real coefficients, its complex roots always come in conjugate pairs. This is because complex roots are symmetric with respect to the real axis, and their conjugates are obtained by changing the sign of the imaginary part.

Real-World Applications

Complex roots and conjugate pairs have numerous real-world applications in various fields, including engineering, physics, and computer science. For example, in electrical engineering, complex roots are used to analyze and design electronic circuits. In physics, complex roots are used to model and analyze the behavior of physical systems, such as oscillators and filters.

Conclusion

In conclusion, complex solutions of quadratic equations are essential in understanding the behavior of quadratic functions and their applications in various fields. The correct statement regarding a given solution is that the other solution to the quadratic function $h$ is $-4 - 7i$. This is because the solutions to a quadratic equation with complex roots always come in conjugate pairs.

Final Thoughts

In this article, we explored the concept of complex solutions of quadratic equations and determined the correct statement regarding a given solution. We also discussed the importance of conjugate pairs in understanding complex roots and their applications in various fields. By understanding complex solutions and conjugate pairs, we can better analyze and design complex systems, which is essential in various fields, including engineering, physics, and computer science.

References

• [1] "Complex Numbers" by Math Open Reference
• [2] "Quadratic Equations" by Khan Academy
• [3] "Complex Roots" by Wolfram MathWorld

Additional Resources

• [1] "Complex Analysis" by MIT OpenCourseWare
• [2] "Quadratic Equations and Functions" by CK-12
• [3] "Complex Numbers and Quadratic Equations" by Purplemath
  Complex Solutions of Quadratic Equations: Q&A =====================================================

Understanding Complex Roots

In our previous article, we explored the concept of complex solutions of quadratic equations and determined the correct statement regarding a given solution. In this article, we will answer some frequently asked questions about complex solutions of quadratic equations.

Q: What are complex numbers?

A: Complex numbers are numbers that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$.

Q: What are the properties of complex numbers?

A: Complex numbers have several properties, including:

• Addition: The sum of two complex numbers is another complex number.
• Multiplication: The product of two complex numbers is another complex number.
• Conjugation: The conjugate of a complex number is obtained by changing the sign of the imaginary part.
• Modulus: The modulus of a complex number is the distance from the origin to the point representing the complex number in the complex plane.

Q: What is the difference between complex roots and conjugate pairs?

A: Complex roots are solutions to a quadratic equation that involve imaginary numbers. Conjugate pairs are pairs of complex roots that are symmetric with respect to the real axis. In other words, if a complex root is $a + bi$, its conjugate pair is $a - bi$.

Q: Why do complex roots always come in conjugate pairs?

A: Complex roots always come in conjugate pairs because the coefficients of the quadratic equation are real numbers. When a quadratic equation has real coefficients, its complex roots are symmetric with respect to the real axis, and their conjugates are obtained by changing the sign of the imaginary part.

Q: How do I find the conjugate of a complex number?

A: To find the conjugate of a complex number, you need to change the sign of the imaginary part. For example, the conjugate of $3 + 4i$ is $3 - 4i$.

Q: What is the significance of complex roots in real-world applications?

A: Complex roots have numerous real-world applications in various fields, including engineering, physics, and computer science. For example, in electrical engineering, complex roots are used to analyze and design electronic circuits. In physics, complex roots are used to model and analyze the behavior of physical systems, such as oscillators and filters.

Q: How do I apply complex roots in real-world problems?

A: To apply complex roots in real-world problems, you need to:

1. Identify the problem: Determine the type of problem you are trying to solve.
2. Determine the complex roots: Find the complex roots of the quadratic equation that represents the problem.
3. Analyze the complex roots: Analyze the complex roots to determine their significance in the problem.
4. Apply the complex roots: Apply the complex roots to solve the problem.

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

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

• Not considering the conjugate pair: Failing to consider the conjugate pair of a complex root can lead to incorrect solutions.
• Not analyzing the complex roots: Failing to analyze the complex roots can lead to incorrect conclusions.
• Not applying the complex roots correctly: Failing to apply the complex roots correctly can lead to incorrect solutions.

Conclusion

In conclusion, complex solutions of quadratic equations are essential in understanding the behavior of quadratic functions and their applications in various fields. By understanding complex roots and conjugate pairs, we can better analyze and design complex systems, which is essential in various fields, including engineering, physics, and computer science.

Final Thoughts

In this article, we answered some frequently asked questions about complex solutions of quadratic equations. We hope that this article has provided you with a better understanding of complex roots and conjugate pairs and their applications in real-world problems.

References

• [1] "Complex Numbers" by Math Open Reference
• [2] "Quadratic Equations" by Khan Academy
• [3] "Complex Roots" by Wolfram MathWorld

Additional Resources

• [1] "Complex Analysis" by MIT OpenCourseWare
• [2] "Quadratic Equations and Functions" by CK-12
• [3] "Complex Numbers and Quadratic Equations" by Purplemath