What Should Be The First Step When Solving The Equation $\sqrt{1-x} - 1 = X$?A. Add 1 To Both Sides Of The Equation.B. Drop The Square Root Symbol To Get $1-x-1=x$.C. Subtract $x$ From Both Sides.D. Square Both Sides Of The

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Introduction

Solving equations is a fundamental concept in mathematics, and it requires a systematic approach to arrive at the correct solution. When faced with an equation involving square roots, it's essential to take the first step carefully to avoid any potential errors. In this article, we will explore the correct first step when solving the equation 1−x−1=x\sqrt{1-x} - 1 = x.

Understanding the Equation

The given equation is 1−x−1=x\sqrt{1-x} - 1 = x. This equation involves a square root, which makes it a bit more complex than a simple linear equation. To solve this equation, we need to isolate the variable xx.

Analyzing the Options

Let's analyze the options provided:

A. Add 1 to both sides of the equation. This option seems like a reasonable approach, but we need to consider the implications of adding 1 to both sides.

B. Drop the square root symbol to get 1−x−1=x1-x-1=x. This option is not a valid approach, as it ignores the square root symbol and changes the equation's structure.

C. Subtract xx from both sides. This option is also not a valid approach, as it doesn't address the square root symbol.

D. Square both sides of the equation. This option seems like a promising approach, but we need to consider the implications of squaring both sides.

The Correct First Step

The correct first step when solving the equation 1−x−1=x\sqrt{1-x} - 1 = x is to add 1 to both sides of the equation. This is because adding 1 to both sides will help us eliminate the negative term and isolate the square root term.

By adding 1 to both sides, we get:

1−x−1+1=x+1\sqrt{1-x} - 1 + 1 = x + 1

This simplifies to:

1−x=x+1\sqrt{1-x} = x + 1

Why Adding 1 is the Correct Approach

Adding 1 to both sides is the correct approach because it helps us eliminate the negative term and isolate the square root term. By doing so, we can then focus on solving the equation without worrying about the negative term.

Squaring Both Sides: A Common Mistake

Squaring both sides of the equation is a common mistake when dealing with equations involving square roots. This approach can lead to incorrect solutions and should be avoided.

When we square both sides of the equation, we get:

(1−x)2=(x+1)2(\sqrt{1-x})^2 = (x + 1)^2

This simplifies to:

1−x=x2+2x+11-x = x^2 + 2x + 1

This equation is more complex than the original equation and can lead to incorrect solutions.

Conclusion

In conclusion, the correct first step when solving the equation 1−x−1=x\sqrt{1-x} - 1 = x is to add 1 to both sides of the equation. This approach helps us eliminate the negative term and isolate the square root term, making it easier to solve the equation. By avoiding common mistakes, such as dropping the square root symbol or squaring both sides, we can arrive at the correct solution.

Additional Tips and Tricks

When dealing with equations involving square roots, it's essential to take the first step carefully. Here are some additional tips and tricks to keep in mind:

  • Isolate the square root term: When dealing with equations involving square roots, it's essential to isolate the square root term. This will help you focus on solving the equation without worrying about the negative term.
  • Avoid squaring both sides: Squaring both sides of the equation can lead to incorrect solutions. Instead, focus on isolating the square root term and then solving the equation.
  • Use algebraic manipulations: Algebraic manipulations, such as adding or subtracting terms, can help you isolate the square root term and solve the equation.

By following these tips and tricks, you can become proficient in solving equations involving square roots and arrive at the correct solution.

Final Thoughts

Q&A: Solving Equations Involving Square Roots

Q: What is the first step when solving the equation 1−x−1=x\sqrt{1-x} - 1 = x? A: The first step when solving the equation 1−x−1=x\sqrt{1-x} - 1 = x is to add 1 to both sides of the equation. This helps to eliminate the negative term and isolate the square root term.

Q: Why is adding 1 to both sides the correct approach? A: Adding 1 to both sides is the correct approach because it helps to eliminate the negative term and isolate the square root term. By doing so, we can then focus on solving the equation without worrying about the negative term.

Q: What is the next step after adding 1 to both sides? A: After adding 1 to both sides, the equation becomes 1−x=x+1\sqrt{1-x} = x + 1. The next step is to square both sides of the equation to eliminate the square root.

Q: Why is squaring both sides the next step? A: Squaring both sides is the next step because it helps to eliminate the square root and isolate the variable xx. By squaring both sides, we can then solve for xx.

Q: What is the equation after squaring both sides? A: After squaring both sides, the equation becomes 1−x=(x+1)21-x = (x + 1)^2. This equation can be simplified to 1−x=x2+2x+11-x = x^2 + 2x + 1.

Q: How do we solve for xx? A: To solve for xx, we need to isolate the variable xx on one side of the equation. We can do this by moving all the terms involving xx to one side of the equation and the constant terms to the other side.

Q: What is the final solution for xx? A: The final solution for xx is x=−3±52x = \frac{-3 \pm \sqrt{5}}{2}. This is the solution to the equation 1−x−1=x\sqrt{1-x} - 1 = x.

Q: What are some common mistakes to avoid when solving equations involving square roots? A: Some common mistakes to avoid when solving equations involving square roots include:

  • Dropping the square root symbol
  • Squaring both sides of the equation
  • Not isolating the square root term
  • Not using algebraic manipulations to solve the equation

Q: How can I practice solving equations involving square roots? A: You can practice solving equations involving square roots by working through examples and exercises. You can also try solving equations involving square roots on your own and then check your solutions with a calculator or a teacher.

Q: What are some real-world applications of solving equations involving square roots? A: Solving equations involving square roots has many real-world applications, including:

  • Physics: Solving equations involving square roots is used to calculate distances, velocities, and accelerations.
  • Engineering: Solving equations involving square roots is used to design and optimize systems.
  • Computer Science: Solving equations involving square roots is used in algorithms and data structures.

Conclusion

Solving equations involving square roots requires a systematic approach. By taking the first step carefully and avoiding common mistakes, you can arrive at the correct solution. Remember to isolate the square root term, avoid squaring both sides, and use algebraic manipulations to solve the equation. With practice and patience, you can become proficient in solving equations involving square roots and tackle even the most complex equations with confidence.