Solve The Following Quadratic Equation For All Values Of $x$ In Simplest Form.$2 - X^2 = -7$

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $2 - x^2 = -7$, and provide a step-by-step guide on how to simplify it.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In our given equation, $2 - x^2 = -7$, we can rewrite it in the standard form as $x^2 - 2 = 7$.

Step 1: Rearrange the Equation

To solve the equation, we need to isolate the variable $x$. We can start by rearranging the equation to get all the terms on one side. In this case, we can add $2$ to both sides of the equation to get:

x2=7+2x^2 = 7 + 2

This simplifies to:

x2=9x^2 = 9

Step 2: Take the Square Root

Now that we have $x^2 = 9$, we can take the square root of both sides to get:

x=±9x = \pm \sqrt{9}

This gives us two possible values for $x$:

x=9x = \sqrt{9}

x=−9x = -\sqrt{9}

Step 3: Simplify the Square Root

The square root of $9$ is $3$, so we can simplify the equation to get:

x=±3x = \pm 3

This means that the solutions to the equation are $x = 3$ and $x = -3$.

Conclusion

In this article, we solved the quadratic equation $2 - x^2 = -7$ and simplified it to get the solutions $x = 3$ and $x = -3$. We followed a step-by-step guide to rearrange the equation, take the square root, and simplify the square root to get the final solutions.

Tips and Tricks

  • When solving quadratic equations, always start by rearranging the equation to get all the terms on one side.
  • Take the square root of both sides to get the possible values for the variable.
  • Simplify the square root to get the final solutions.

Real-World Applications

Quadratic equations have many real-world applications, such as:

  • Modeling the trajectory of a projectile
  • Calculating the area of a circle
  • Determining the maximum or minimum value of a function

Common Mistakes

  • Not rearranging the equation to get all the terms on one side
  • Not taking the square root of both sides
  • Not simplifying the square root

Practice Problems

Try solving the following quadratic equations:

  • x2+4=0x^2 + 4 = 0

  • x2−6=0x^2 - 6 = 0

  • x2+2x+1=0x^2 + 2x + 1 = 0

Conclusion

Introduction

Quadratic equations can be a challenging topic for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this important concept.

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 (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to follow these steps:

  1. Rearrange the equation to get all the terms on one side.
  2. Take the square root of both sides to get the possible values for the variable.
  3. Simplify the square root to get the final solutions.

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

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (in this case, $x$) is one. The general form of a linear equation is $ax + b = 0$, where $a$ and $b$ are constants. Quadratic equations, on the other hand, have a degree of two, making them more complex and challenging to solve.

Q: Can I use a calculator to solve quadratic equations?

A: Yes, you can use a calculator to solve quadratic equations. However, it's essential to understand the underlying concepts and formulas to ensure you're using the calculator correctly.

Q: What is the formula for solving quadratic equations?

A: The formula for solving quadratic equations is:

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

This formula is known as the quadratic formula, and it can be used to solve any quadratic equation.

Q: What are the different types of solutions for quadratic equations?

A: Quadratic equations can have different types of solutions, including:

  • Real and distinct solutions: These are solutions that are real numbers and are distinct from each other.
  • Real and repeated solutions: These are solutions that are real numbers and are repeated.
  • Complex solutions: These are solutions that are complex numbers, which include imaginary numbers.

Q: Can I use quadratic equations to model real-world problems?

A: Yes, quadratic equations can be used to model real-world problems, such as:

  • Modeling the trajectory of a projectile
  • Calculating the area of a circle
  • Determining the maximum or minimum value of a function

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

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

  • Not rearranging the equation to get all the terms on one side
  • Not taking the square root of both sides
  • Not simplifying the square root

Conclusion

Quadratic equations can be a challenging topic, but with practice and understanding, you can become proficient in solving them. Remember to always follow the steps outlined in this article, and don't be afraid to ask for help if you need it. With time and practice, you'll become a master of quadratic equations.

Practice Problems

Try solving the following quadratic equations:

  • x2+4=0x^2 + 4 = 0

  • x2−6=0x^2 - 6 = 0

  • x2+2x+1=0x^2 + 2x + 1 = 0

Additional Resources

For more information on quadratic equations, check out the following resources:

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equations
  • Wolfram Alpha: Quadratic Equations

Conclusion

We hope this article has helped you better understand quadratic equations and how to solve them. Remember to practice regularly and seek help when needed. With time and practice, you'll become a master of quadratic equations and be able to tackle even the most challenging problems.