What Are The Solutions Of $x^2 - 4 = 0$?A. $x = -4$ Or $x = 4$ B. $x = -2$ Or $x = 2$ C. $x = 2$ D. $x = 4$

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. 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. In this article, we will focus on solving the quadratic equation $x^2 - 4 = 0$.

Understanding the Quadratic Equation

The quadratic equation $x^2 - 4 = 0$ can be rewritten as $x^2 = 4$. This equation represents a parabola that opens upwards, and its graph is a U-shaped curve. The equation has two solutions, which are the values of $x$ that satisfy the equation.

Factoring the Quadratic Equation

One of the methods to solve a quadratic equation is by factoring. Factoring involves expressing the quadratic equation as a product of two binomials. In this case, we can factor the quadratic equation $x^2 - 4 = 0$ as follows:

x2−4=(x−2)(x+2)=0x^2 - 4 = (x - 2)(x + 2) = 0

Using the Zero Product Property

The zero product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In this case, we have:

(x−2)(x+2)=0(x - 2)(x + 2) = 0

Using the zero product property, we can set each factor equal to zero and solve for $x$:

x−2=0⇒x=2x - 2 = 0 \Rightarrow x = 2

x+2=0⇒x=−2x + 2 = 0 \Rightarrow x = -2

Conclusion

In conclusion, the solutions of the quadratic equation $x^2 - 4 = 0$ are $x = 2$ and $x = -2$. These solutions can be obtained by factoring the quadratic equation and using the zero product property. The quadratic equation $x^2 - 4 = 0$ represents a parabola that opens upwards, and its graph is a U-shaped curve.

Final Answer

The final answer is: B\boxed{B}

Discussion

The quadratic equation $x^2 - 4 = 0$ is a simple equation that can be solved using various methods such as factoring and the zero product property. The solutions of the equation are $x = 2$ and $x = -2$. These solutions can be obtained by factoring the quadratic equation and using the zero product property.

Related Topics

  • Quadratic equations
  • Factoring
  • Zero product property
  • Parabolas
  • U-shaped curves

References

  • [1] "Quadratic Equations" by Math Open Reference
  • [2] "Factoring Quadratic Equations" by Purplemath
  • [3] "Zero Product Property" by Math Is Fun

Additional Resources

  • [1] "Quadratic Equations" by Khan Academy
  • [2] "Factoring Quadratic Equations" by IXL
  • [3] "Zero Product Property" by CK-12

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we discussed the solutions of the quadratic equation $x^2 - 4 = 0$. In this article, we will answer some frequently asked questions (FAQs) about quadratic equations.

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. 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: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The method you choose will depend on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve a quadratic equation. The formula is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared variable, while a linear equation does not.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. This is because the graph of a quadratic equation is a parabola, and a parabola can intersect the x-axis at most two times.

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

A: To determine the number of solutions of a quadratic equation, you can use the discriminant, which is the expression $b^2 - 4ac$ under the square root in the quadratic formula. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the discriminant?

A: The discriminant is the expression $b^2 - 4ac$ under the square root in the quadratic formula. It is used to determine the number of solutions of a quadratic equation.

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. If the discriminant is negative, the equation has no real solutions, but it may have complex solutions.

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

A: To find the complex solutions of a quadratic equation, you can use the quadratic formula and take the square root of the negative discriminant. This will give you two complex solutions.

Q: What is the difference between a quadratic equation and a polynomial equation of degree three?

A: A quadratic equation is a polynomial equation of degree two, while a polynomial equation of degree three is a cubic equation. A cubic equation has a cubed variable, while a quadratic equation has a squared variable.

Q: Can a quadratic equation be used to model real-world problems?

A: Yes, a quadratic equation can be used to model real-world problems. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation.

Q: How do I use a quadratic equation to model a real-world problem?

A: To use a quadratic equation to model a real-world problem, you need to identify the variables and the relationships between them. You can then use the quadratic equation to describe the behavior of the system.

Q: What are some common applications of quadratic equations?

A: Quadratic equations have many common applications, including physics, engineering, economics, and computer science. They are used to model the behavior of systems, optimize functions, and solve problems.

Q: How do I choose the right method to solve a quadratic equation?

A: To choose the right method to solve a quadratic equation, you need to consider the specific equation and your personal preference. You can use factoring, the quadratic formula, or completing the square, depending on the equation and your skills.

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 the solutions for extraneous solutions
  • Not using the correct method to solve the equation
  • Not simplifying the solutions
  • Not checking the solutions for complex solutions

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solutions back into the original equation and check if they are true. If they are not true, then they are extraneous solutions.

Q: What are some common tools and software used to solve quadratic equations?

A: Some common tools and software used to solve quadratic equations include:

  • Graphing calculators
  • Computer algebra systems (CAS)
  • Online calculators
  • Spreadsheets

Q: How do I use a graphing calculator to solve a quadratic equation?

A: To use a graphing calculator to solve a quadratic equation, you need to enter the equation into the calculator and use the solve function. The calculator will then give you the solutions to the equation.

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

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

  • Textbooks
  • Online tutorials
  • Video lectures
  • Practice problems
  • Online communities

Q: How do I practice solving quadratic equations?

A: To practice solving quadratic equations, you need to work on a variety of problems, including factoring, using the quadratic formula, and completing the square. You can use practice problems, online resources, and worksheets to help you practice.

Q: What are some common challenges when solving quadratic equations?

A: Some common challenges when solving quadratic equations include:

  • Not understanding the concept of quadratic equations
  • Not being able to factor the equation
  • Not being able to use the quadratic formula
  • Not being able to complete the square

Q: How do I overcome these challenges?

A: To overcome these challenges, you need to practice solving quadratic equations, work on your problem-solving skills, and seek help from a teacher or tutor if needed.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we have answered some frequently asked questions (FAQs) about quadratic equations, including how to solve them, how to determine the number of solutions, and how to use them to model real-world problems. We hope that this article has been helpful in answering your questions and providing you with a better understanding of quadratic equations.