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

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding their solutions is crucial for various applications in science, engineering, and other fields. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on a specific quadratic function and explore the nature of its solutions.

The Given Solution

The given solution to the quadratic function is ${-4+7i}$. This is a complex number, which means it has both real and imaginary parts. In this case, the real part is ${-4}$ and the imaginary part is ${7}$.

Understanding Quadratic Equations

A quadratic equation can be written in the form ${ax^2 + bx + c = 0}$, where ${a}$, ${b}$, and ${c}$ are constants, and ${x}$ is the variable. The solutions to this equation are the values of ${x}$ that satisfy the equation.

The Nature of Solutions

When solving a quadratic equation, we can have one of three possible scenarios:

• Two distinct real solutions: In this case, the equation has two distinct real solutions, which means the solutions are not equal.
• One repeated real solution: In this case, the equation has one repeated real solution, which means the solutions are equal.
• Two complex solutions: In this case, the equation has two complex solutions, which means the solutions are complex numbers.

Analyzing the Given Solution

Given that the solution is ${-4+7i}$, we can conclude that the quadratic function has two complex solutions. This is because the solution is a complex number, which means it has both real and imaginary parts.

The Other Solution

Since the given solution is ${-4+7i}$, we can find the other solution by using the fact that the sum of the solutions is equal to the negation of the coefficient of the linear term, divided by the coefficient of the quadratic term. In this case, the coefficient of the linear term is ${b}$ and the coefficient of the quadratic term is ${a}$.

The Correct Statement

Based on the analysis above, we can conclude that the correct statement is:

• The other solution to function ${h}$ is ${-4-7i}$.

This is because the sum of the solutions is equal to the negation of the coefficient of the linear term, divided by the coefficient of the quadratic term. In this case, the coefficient of the linear term is ${b}$ and the coefficient of the quadratic term is ${a}$.

Conclusion

In conclusion, the given solution ${-4+7i}$ is a complex number, which means the quadratic function has two complex solutions. The other solution can be found by using the fact that the sum of the solutions is equal to the negation of the coefficient of the linear term, divided by the coefficient of the quadratic term. Therefore, the correct statement is that the other solution to function ${h}$ is ${-4-7i}$.

References

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

Additional Resources

• [1] "Quadratic Formula" by Wolfram MathWorld
• [2] "Complex Number Calculator" by Calculator Soup

Discussion

What are some common applications of quadratic equations in real-life scenarios?

• Engineering: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Physics: Quadratic equations are used to describe the motion of particles in a magnetic field.
• Computer Science: Quadratic equations are used in algorithms for solving systems of linear equations.

What are some common mistakes to avoid when solving quadratic equations?

• Not checking for complex solutions: Quadratic equations can have complex solutions, which are often overlooked.
• Not using the correct formula: The quadratic formula is a powerful tool for solving quadratic equations, but it must be used correctly.
• Not checking for repeated solutions: Quadratic equations can have repeated solutions, which must be checked for.

How can you apply the concepts learned in this article to real-life scenarios?

• Modeling motion: Quadratic equations can be used to model the motion of objects under the influence of gravity, friction, and other forces.
• Solving systems of equations: Quadratic equations can be used to solve systems of linear equations.
• Analyzing data: Quadratic equations can be used to analyze data and make predictions.

Final Thoughts

$$$$

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It can be written in the form ${ax^2 + bx + c = 0}$, where ${a}$, ${b}$, and ${c}$ are constants, and ${x}$ is the variable.

Q: What are the different types of solutions to a quadratic equation?

A: There are three possible scenarios for the solutions to a quadratic equation:

• Two distinct real solutions: In this case, the equation has two distinct real solutions, which means the solutions are not equal.
• One repeated real solution: In this case, the equation has one repeated real solution, which means the solutions are equal.
• Two complex solutions: In this case, the equation has two complex solutions, which means the solutions are complex numbers.

Q: How do I find the solutions to a quadratic equation?

A: To find the solutions to a quadratic equation, you can use the quadratic formula:

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

This formula will give you two possible solutions, which you can then check to see if they are real or complex.

Q: What is the significance of the discriminant in a quadratic equation?

A: The discriminant is the expression ${b^2 - 4ac}$ under the square root in the quadratic formula. It determines the nature of the solutions to the equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one repeated real solution. If the discriminant is negative, the equation has two complex solutions.

Q: How do I apply the concepts of quadratic equations to real-life scenarios?

A: Quadratic equations have many applications in science, engineering, and other fields. Some examples include:

• Modeling motion: Quadratic equations can be used to model the motion of objects under the influence of gravity, friction, and other forces.
• Solving systems of equations: Quadratic equations can be used to solve systems of linear equations.
• Analyzing data: Quadratic equations can be used to analyze data and make predictions.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Not checking for complex solutions: Quadratic equations can have complex solutions, which are often overlooked.
• Not using the correct formula: The quadratic formula is a powerful tool for solving quadratic equations, but it must be used correctly.
• Not checking for repeated solutions: Quadratic equations can have repeated solutions, which must be checked for.

Q: How can I use quadratic equations to solve problems in my field?

A: Quadratic equations have many applications in various fields, including science, engineering, and economics. To use quadratic equations to solve problems in your field, you can:

• Model real-world phenomena: Quadratic equations can be used to model real-world phenomena, such as the motion of objects or the growth of populations.
• Solve systems of equations: Quadratic equations can be used to solve systems of linear equations, which is useful in many fields, including engineering and economics.
• Analyze data: Quadratic equations can be used to analyze data and make predictions, which is useful in many fields, including science and economics.

Q: What are some resources for learning more about quadratic equations?

A: Some resources for learning more about quadratic equations include:

• Textbooks: There are many textbooks available that cover quadratic equations in detail.
• Online resources: There are many online resources available that provide tutorials and examples on quadratic equations.
• Mathematical software: Mathematical software, such as Mathematica or Maple, can be used to solve quadratic equations and visualize their solutions.

Q: How can I apply the concepts of quadratic equations to my own life?

A: Quadratic equations have many applications in real-life scenarios, including:

• Modeling personal finance: Quadratic equations can be used to model personal finance, such as saving for retirement or investing in stocks.
• Analyzing data: Quadratic equations can be used to analyze data and make predictions, which is useful in many areas of life, including business and personal finance.
• Solving problems: Quadratic equations can be used to solve problems in many areas of life, including science, engineering, and economics.

Q: What are some common applications of quadratic equations in science and engineering?

A: Quadratic equations have many applications in science and engineering, including:

• Modeling motion: Quadratic equations can be used to model the motion of objects under the influence of gravity, friction, and other forces.
• Solving systems of equations: Quadratic equations can be used to solve systems of linear equations, which is useful in many areas of science and engineering.
• Analyzing data: Quadratic equations can be used to analyze data and make predictions, which is useful in many areas of science and engineering.

Q: How can I use quadratic equations to solve problems in my own field?

A: To use quadratic equations to solve problems in your own field, you can:

• Model real-world phenomena: Quadratic equations can be used to model real-world phenomena, such as the motion of objects or the growth of populations.
• Solve systems of equations: Quadratic equations can be used to solve systems of linear equations, which is useful in many fields, including engineering and economics.
• Analyze data: Quadratic equations can be used to analyze data and make predictions, which is useful in many fields, including science and economics.

Q: What are some common mistakes to avoid when applying quadratic equations to real-life scenarios?

A: Some common mistakes to avoid when applying quadratic equations to real-life scenarios include:

• Not checking for complex solutions: Quadratic equations can have complex solutions, which are often overlooked.
• Not using the correct formula: The quadratic formula is a powerful tool for solving quadratic equations, but it must be used correctly.
• Not checking for repeated solutions: Quadratic equations can have repeated solutions, which must be checked for.

Q: How can I use quadratic equations to make predictions in my own field?

A: To use quadratic equations to make predictions in your own field, you can:

• Model real-world phenomena: Quadratic equations can be used to model real-world phenomena, such as the motion of objects or the growth of populations.
• Solve systems of equations: Quadratic equations can be used to solve systems of linear equations, which is useful in many fields, including engineering and economics.
• Analyze data: Quadratic equations can be used to analyze data and make predictions, which is useful in many fields, including science and economics.

Q: What are some common applications of quadratic equations in economics?

A: Quadratic equations have many applications in economics, including:

• Modeling economic growth: Quadratic equations can be used to model economic growth, such as the growth of GDP or the growth of population.
• Solving systems of equations: Quadratic equations can be used to solve systems of linear equations, which is useful in many areas of economics, including macroeconomics and microeconomics.
• Analyzing data: Quadratic equations can be used to analyze data and make predictions, which is useful in many areas of economics, including finance and marketing.

Q: How can I use quadratic equations to solve problems in my own field of economics?

A: To use quadratic equations to solve problems in your own field of economics, you can:

• Model real-world phenomena: Quadratic equations can be used to model real-world phenomena, such as the growth of GDP or the growth of population.
• Solve systems of equations: Quadratic equations can be used to solve systems of linear equations, which is useful in many areas of economics, including macroeconomics and microeconomics.
• Analyze data: Quadratic equations can be used to analyze data and make predictions, which is useful in many areas of economics, including finance and marketing.

Q: What are some common mistakes to avoid when applying quadratic equations to economic problems?

A: Some common mistakes to avoid when applying quadratic equations to economic problems include:

• Not checking for complex solutions: Quadratic equations can have complex solutions, which are often overlooked.
• Not using the correct formula: The quadratic formula is a powerful tool for solving quadratic equations, but it must be used correctly.
• Not checking for repeated solutions: Quadratic equations can have repeated solutions, which must be checked for.

Q: How can I use quadratic equations to make predictions in my own field of economics?

A: To use quadratic equations to make predictions in your own field of economics, you can:

• Model real-world phenomena: Quadratic equations can be used to model real-world phenomena, such as the growth of GDP or the growth of population.
• Solve systems of equations: Quadratic equations can be used to solve systems of linear equations, which is useful in many areas of economics, including macroeconomics and microeconomics.
• Analyze data: Quadratic equations can be used to analyze data and make predictions, which is useful in many areas of economics, including finance and marketing.