Solve For { X $} : : : { \sqrt{2x + 66} - 19 = -11 \}

Mar 13, 2025 by ADMIN 54 views

**Solving Equations with Square Roots: A Step-by-Step Guide**

Introduction

Solving equations with square roots can be a challenging task, but with the right approach, it can be made easier. In this article, we will focus on solving equations that involve square roots, specifically the equation $\sqrt{2x + 66} - 19 = -11$ . We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Equation

The given equation is $\sqrt{2x + 66} - 19 = -11$ . To solve this equation, we need to isolate the square root term. The first step is to add 19 to both sides of the equation to get rid of the negative term.

\sqrt{2x + 66} - 19 + 19 = -11 + 19

This simplifies to:

\sqrt{2x + 66} = 8

Isolating the Square Root Term

Now that we have isolated the square root term, we can square both sides of the equation to get rid of the square root. This will give us an equation that we can solve for x.

(\sqrt{2x + 66})^2 = 8^2

This simplifies to:

2x + 66 = 64

Solving for x

Now that we have a linear equation, we can solve for x by isolating the variable. The first step is to subtract 66 from both sides of the equation to get rid of the constant term.

2x + 66 - 66 = 64 - 66

This simplifies to:

2x = -2

Final Step

The final step is to divide both sides of the equation by 2 to solve for x.

\frac{2x}{2} = \frac{-2}{2}

This simplifies to:

x = -1

Conclusion

Tips and Tricks

When solving equations with square roots, it's essential to isolate the square root term before squaring both sides of the equation.
Make sure to check your work by plugging the solution back into the original equation.
If you're struggling to solve an equation with square roots, try breaking it down into smaller steps and using algebraic manipulations to simplify the equation.

Common Mistakes

Failing to isolate the square root term before squaring both sides of the equation.
Not checking the solution by plugging it back into the original equation.
Not using algebraic manipulations to simplify the equation.

Real-World Applications

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

Physics: Solving equations with square roots is essential in physics, particularly in the study of motion and energy.
Engineering: Solving equations with square roots is critical in engineering, particularly in the design of structures and systems.
Computer Science: Solving equations with square roots is used in computer science, particularly in the development of algorithms and data structures.

Conclusion

Introduction

Solving equations with square roots can be a challenging task, but with the right approach, it can be made easier. In this article, we will provide a Q&A guide to help you understand and solve equations with square roots.

Q: What is a square root?

A: A square root is a number that, when multiplied by itself, gives a specified value. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

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

A: To solve an equation with a square root, you need to isolate the square root term and then square both sides of the equation. This will give you a linear equation that you can solve for x.

Q: What is the first step in solving an equation with a square root?

A: The first step in solving an equation with a square root is to isolate the square root term. This means getting the square root term by itself on one side of the equation.

Q: How do I isolate the square root term?

A: To isolate the square root term, you need to add or subtract the same value from both sides of the equation. For example, if the equation is $\sqrt{2x + 66} - 19 = -11$ , you can add 19 to both sides of the equation to get $\sqrt{2x + 66} = 8$ .

Q: What is the next step in solving an equation with a square root?

A: The next step in solving an equation with a square root is to square both sides of the equation. This will give you a linear equation that you can solve for x.

Q: How do I square both sides of the equation?

A: To square both sides of the equation, you need to multiply both sides of the equation by themselves. For example, if the equation is $\sqrt{2x + 66} = 8$ , you can square both sides of the equation to get $2x + 66 = 64$ .

Q: What is the final step in solving an equation with a square root?

A: The final step in solving an equation with a square root is to solve for x. This means isolating the variable x and finding its value.

Q: How do I solve for x?

A: To solve for x, you need to isolate the variable x and find its value. This means getting x by itself on one side of the equation.

Q: What are some common mistakes to avoid when solving equations with square roots?

A: Some common mistakes to avoid when solving equations with square roots include:

Failing to isolate the square root term before squaring both sides of the equation.
Not checking the solution by plugging it back into the original equation.
Not using algebraic manipulations to simplify the equation.

Q: What are some real-world applications of solving equations with square roots?

A: Some real-world applications of solving equations with square roots include:

Physics: Solving equations with square roots is essential in physics, particularly in the study of motion and energy.
Engineering: Solving equations with square roots is critical in engineering, particularly in the design of structures and systems.
Computer Science: Solving equations with square roots is used in computer science, particularly in the development of algorithms and data structures.

Conclusion

Solving equations with square roots requires a step-by-step approach. By isolating the square root term and squaring both sides of the equation, we can solve for x. In this article, we provided a Q&A guide to help you understand and solve equations with square roots. With practice and patience, you can master the art of solving equations with square roots and apply it to real-world problems.