Solve Each Equation By Taking Square Roots.13) 4 X 2 − 5 = − 80 4x^2 - 5 = -80 4 X 2 − 5 = − 80

Introduction

In this article, we will focus on solving equations that involve square roots. Taking square roots is an essential concept in mathematics, and it is used to solve various types of equations. In this discussion, we will explore how to solve equations by taking square roots, with a focus on the given equation $4x^2 - 5 = -80$.

Understanding the Concept of Square Roots

Before we dive into solving the equation, let's briefly review the concept of square roots. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. The square root of a number is denoted by the symbol $\sqrt{}$. In this article, we will use the square root symbol to represent the square root of a number.

Solving the Equation

Now that we have a basic understanding of square roots, let's move on to solving the equation $4x^2 - 5 = -80$. To solve this equation, we need to isolate the variable $x$. We can start by adding 5 to both sides of the equation, which gives us:

$$4x^2 = -80 + 5$$

$$4x^2 = -75$$

Next, we can divide both sides of the equation by 4, which gives us:

$$x^2 = \frac{-75}{4}$$

$$x^2 = -18.75$$

Now, we can take the square root of both sides of the equation, which gives us:

$$x = \pm \sqrt{-18.75}$$

Simplifying the Square Root

The square root of a negative number is an imaginary number, which is denoted by the symbol $i$. In this case, we can simplify the square root of -18.75 as follows:

$$\sqrt{-18.75} = \sqrt{-1} \times \sqrt{18.75}$$

$$\sqrt{-18.75} = i \times \sqrt{18.75}$$

$$\sqrt{-18.75} = i \times 4.33$$

$$\sqrt{-18.75} = 4.33i$$

Solving for x

Now that we have simplified the square root, we can solve for $x$:

$$x = \pm 4.33i$$

Conclusion

In this article, we have solved the equation $4x^2 - 5 = -80$ by taking square roots. We started by adding 5 to both sides of the equation, then divided both sides by 4, and finally took the square root of both sides. We simplified the square root by expressing it as an imaginary number, and then solved for $x$. The final solution is $x = \pm 4.33i$.

Tips and Tricks

Here are some tips and tricks to keep in mind when solving equations by taking square roots:

• Make sure to simplify the square root by expressing it as an imaginary number.
• Use the correct notation for imaginary numbers, which is $i$.
• Be careful when solving for $x$, as the solution may be an imaginary number.

Common Mistakes

Here are some common mistakes to avoid when solving equations by taking square roots:

• Failing to simplify the square root.
• Using the wrong notation for imaginary numbers.
• Not being careful when solving for $x$.

Real-World Applications

Solving equations by taking square roots has many real-world applications, including:

• Physics: Solving equations involving square roots is essential in physics, particularly in the study of motion and energy.
• Engineering: Engineers use square roots to solve equations involving stress, strain, and other physical quantities.
• Computer Science: Computer scientists use square roots to solve equations involving algorithms and data structures.

Final Thoughts

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Introduction

In our previous article, we explored how to solve equations by taking square roots. We covered the basics of square roots, how to simplify them, and how to solve for the variable $x$. In this article, we will answer some frequently asked questions about solving equations by taking square roots.

Q&A

Q: What is the difference between a square root and a square?

A: A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. A square, on the other hand, is the result of multiplying a number by itself. For example, the square of 4 is 16.

Q: How do I simplify a square root?

A: To simplify a square root, you need to express it as a product of a perfect square and a remaining factor. For example, the square root of 18 can be simplified as follows:

$$\sqrt{18} = \sqrt{9 \times 2}$$

$$\sqrt{18} = 3\sqrt{2}$$

Q: What is an imaginary number?

A: An imaginary number is a complex number that, when squared, gives a negative result. For example, the square of $i$ is $-1$, where $i$ is the imaginary unit.

Q: How do I solve an equation involving a square root?

A: To solve an equation involving a square root, you need to isolate the variable $x$ by adding, subtracting, multiplying, or dividing both sides of the equation by the same value. For example, to solve the equation $x^2 = 16$, you can take the square root of both sides:

$$x = \pm \sqrt{16}$$

$$x = \pm 4$$

Q: What is the difference between a positive and negative square root?

A: A positive square root is a value that, when multiplied by itself, gives the original number. A negative square root is a value that, when multiplied by itself, gives the negative of the original number. For example, the positive square root of 16 is 4, while the negative square root of 16 is -4.

Q: How do I know which square root to use?

A: To determine which square root to use, you need to consider the context of the problem. If the problem involves a physical quantity, such as distance or time, you should use the positive square root. If the problem involves a mathematical quantity, such as a function or a graph, you can use either the positive or negative square root.

Common Mistakes

Here are some common mistakes to avoid when solving equations by taking square roots:

• Failing to simplify the square root.
• Using the wrong notation for imaginary numbers.
• Not being careful when solving for $x$.
• Ignoring the negative square root.

Real-World Applications

Solving equations by taking square roots has many real-world applications, including:

• Physics: Solving equations involving square roots is essential in physics, particularly in the study of motion and energy.
• Engineering: Engineers use square roots to solve equations involving stress, strain, and other physical quantities.
• Computer Science: Computer scientists use square roots to solve equations involving algorithms and data structures.

Final Thoughts

In conclusion, solving equations by taking square roots is an essential concept in mathematics. By following the steps outlined in this article, you can solve equations involving square roots with ease. Remember to simplify the square root, use the correct notation for imaginary numbers, and be careful when solving for $x$. With practice and patience, you will become proficient in solving equations by taking square roots.

Additional Resources

For more information on solving equations by taking square roots, check out the following resources:

• Khan Academy: Solving Equations with Square Roots
• Mathway: Solving Equations with Square Roots
• Wolfram Alpha: Solving Equations with Square Roots

Conclusion

In this article, we have answered some frequently asked questions about solving equations by taking square roots. We have covered the basics of square roots, how to simplify them, and how to solve for the variable $x$. We have also discussed common mistakes to avoid and real-world applications of solving equations by taking square roots. With practice and patience, you will become proficient in solving equations by taking square roots.