Select The Correct Answer.The Complex Solutions Of Which Equation Have A Real Component Of 4?A. X 2 + 4 X + 4 = − 25 X^2 + 4x + 4 = -25 X 2 + 4 X + 4 = − 25 B. X 2 + 8 X + 16 = − 21 X^2 + 8x + 16 = -21 X 2 + 8 X + 16 = − 21 C. X 2 − 4 X + 4 = − 25 X^2 - 4x + 4 = -25 X 2 − 4 X + 4 = − 25 D. X 2 − 8 X + 16 = − 21 X^2 - 8x + 16 = -21 X 2 − 8 X + 16 = − 21

by ADMIN 355 views

Understanding Complex Solutions

In mathematics, a complex solution to a quadratic equation is a value that, when substituted into the equation, results in a true statement. Complex solutions can be expressed in the form of a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i^2 = -1. In this article, we will explore the complex solutions of quadratic equations and determine which equation has a real component of 4.

The Nature of Complex Solutions

Complex solutions arise when the discriminant of a quadratic equation is negative. The discriminant is given by the formula b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. When the discriminant is negative, the quadratic equation has two complex solutions.

The Real Component of Complex Solutions

The real component of a complex solution is the part of the solution that is not imaginary. In other words, it is the part of the solution that is a real number. To determine the real component of a complex solution, we can use the formula x = (-b ± √(b^2 - 4ac)) / 2a, where x is the complex solution.

Analyzing the Options

Let's analyze the options given in the problem:

Option A: x2+4x+4=−25x^2 + 4x + 4 = -25

To determine the complex solutions of this equation, we can start by subtracting -25 from both sides of the equation:

x2+4x+29=0x^2 + 4x + 29 = 0

The discriminant of this equation is b^2 - 4ac = 4^2 - 4(1)(29) = 16 - 116 = -100, which is negative. Therefore, this equation has two complex solutions.

Option B: x2+8x+16=−21x^2 + 8x + 16 = -21

To determine the complex solutions of this equation, we can start by subtracting -21 from both sides of the equation:

x2+8x+37=0x^2 + 8x + 37 = 0

The discriminant of this equation is b^2 - 4ac = 8^2 - 4(1)(37) = 64 - 148 = -84, which is negative. Therefore, this equation has two complex solutions.

Option C: x2−4x+4=−25x^2 - 4x + 4 = -25

To determine the complex solutions of this equation, we can start by subtracting -25 from both sides of the equation:

x2−4x+29=0x^2 - 4x + 29 = 0

The discriminant of this equation is b^2 - 4ac = (-4)^2 - 4(1)(29) = 16 - 116 = -100, which is negative. Therefore, this equation has two complex solutions.

Option D: x2−8x+16=−21x^2 - 8x + 16 = -21

To determine the complex solutions of this equation, we can start by subtracting -21 from both sides of the equation:

x2−8x+37=0x^2 - 8x + 37 = 0

The discriminant of this equation is b^2 - 4ac = (-8)^2 - 4(1)(37) = 64 - 148 = -84, which is negative. Therefore, this equation has two complex solutions.

Determining the Real Component

To determine the real component of the complex solutions, we can use the formula x = (-b ± √(b^2 - 4ac)) / 2a. Let's apply this formula to each of the options:

Option A: x2+4x+4=−25x^2 + 4x + 4 = -25

The complex solutions of this equation are x = (-4 ± √(-100)) / 2 = (-4 ± 10i) / 2 = -2 ± 5i. The real component of these solutions is -2.

Option B: x2+8x+16=−21x^2 + 8x + 16 = -21

The complex solutions of this equation are x = (-8 ± √(-84)) / 2 = (-8 ± 2√21i) / 2 = -4 ± √21i. The real component of these solutions is -4.

Option C: x2−4x+4=−25x^2 - 4x + 4 = -25

The complex solutions of this equation are x = (4 ± √(-100)) / 2 = (4 ± 10i) / 2 = 2 ± 5i. The real component of these solutions is 2.

Option D: x2−8x+16=−21x^2 - 8x + 16 = -21

The complex solutions of this equation are x = (8 ± √(-84)) / 2 = (8 ± 2√21i) / 2 = 4 ± √21i. The real component of these solutions is 4.

Conclusion

Based on the analysis above, we can conclude that the complex solutions of the equation x2−4x+4=−25x^2 - 4x + 4 = -25 have a real component of 2.

Final Answer

Frequently Asked Questions

In the previous article, we explored the complex solutions of quadratic equations and determined which equation has a real component of 4. In this article, we will answer some frequently asked questions related to complex solutions of quadratic equations.

Q: What is the difference between a complex solution and a real solution?

A: A complex solution is a value that, when substituted into the equation, results in a true statement. It can be expressed in the form of a + bi, where a and b are real numbers and i is the imaginary unit. A real solution, on the other hand, is a value that, when substituted into the equation, results in a true statement and is not imaginary.

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

A: To determine the complex solutions of a quadratic equation, you can use the formula x = (-b ± √(b^2 - 4ac)) / 2a, where x is the complex solution.

Q: What is the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is given by the formula b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. If the discriminant is negative, the quadratic equation has two complex solutions.

Q: How do I determine the real component of a complex solution?

A: To determine the real component of a complex solution, you can use the formula x = (-b ± √(b^2 - 4ac)) / 2a. The real component is the part of the solution that is not imaginary.

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

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

Q: How do I know if a quadratic equation has complex solutions?

A: To determine if a quadratic equation has complex solutions, you can check the discriminant. If the discriminant is negative, the quadratic equation has two complex solutions.

Q: Can I use the quadratic formula to find complex solutions?

A: Yes, you can use the quadratic formula to find complex solutions. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.

Q: What is the significance of complex solutions in mathematics?

A: Complex solutions are significant in mathematics because they help us understand the behavior of quadratic equations. They also have applications in fields such as physics, engineering, and computer science.

Q: Can I use complex solutions to solve real-world problems?

A: Yes, complex solutions can be used to solve real-world problems. For example, in physics, complex solutions can be used to model the behavior of oscillating systems.

Conclusion

In this article, we answered some frequently asked questions related to complex solutions of quadratic equations. We hope that this article has provided you with a better understanding of complex solutions and their significance in mathematics.

Final Answer

The final answer is that complex solutions are an important concept in mathematics that can be used to solve quadratic equations and have applications in various fields.

Additional Resources

For more information on complex solutions of quadratic equations, you can refer to the following resources:

We hope that this article has been helpful in understanding complex solutions of quadratic equations. If you have any further questions, please don't hesitate to ask.