Solve For \[$x\$\] In The Equation:$\[ X = 2 \sqrt{x+1} - 1 \\]

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Introduction

In this article, we will delve into solving for x in the given equation: $x = 2 \sqrt{x+1} - 1$. This equation involves a square root and a variable within the square root, making it a bit more complex than a standard linear equation. We will use algebraic manipulation and properties of square roots to isolate x and find its value.

Understanding the Equation

The given equation is $x = 2 \sqrt{x+1} - 1$. To start solving for x, we need to understand the properties of square roots and how they interact with variables. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we have $\sqrt{x+1}$, which means we are looking for a value that, when squared, gives $x+1$.

Step 1: Isolate the Square Root

To solve for x, we need to isolate the square root term. We can start by adding 1 to both sides of the equation:

$x + 1 = 2 \sqrt{x+1}$

This step helps us to get rid of the constant term on the right-hand side and isolate the square root term.

Step 2: Square Both Sides

Now that we have isolated the square root term, we can square both sides of the equation to eliminate the square root:

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

Squaring both sides gives us:

$x^2 + 2x + 1 = 4(x+1)$

Step 3: Expand and Simplify

We can now expand and simplify the equation:

$x^2 + 2x + 1 = 4x + 4$

Subtracting $4x$ from both sides gives us:

$x^2 - 2x - 3 = 0$

Step 4: Factor the Quadratic

The equation $x^2 - 2x - 3 = 0$ is a quadratic equation, and we can factor it:

$(x - 3)(x + 1) = 0$

Step 5: Solve for x

Now that we have factored the quadratic, we can solve for x:

$x - 3 = 0$ or $x + 1 = 0$

Solving for x gives us:

$x = 3$ or $x = -1$

Checking the Solutions

We need to check our solutions to make sure they satisfy the original equation. Plugging $x = 3$ into the original equation gives us:

$3 = 2 \sqrt{3+1} - 1$

Simplifying the right-hand side gives us:

$3 = 2 \sqrt{4} - 1$

$3 = 2(2) - 1$

$3 = 4 - 1$

$3 = 3$

This shows that $x = 3$ is a valid solution.

Plugging $x = -1$ into the original equation gives us:

$-1 = 2 \sqrt{-1+1} - 1$

Simplifying the right-hand side gives us:

$-1 = 2 \sqrt{0} - 1$

$-1 = 2(0) - 1$

$-1 = -1$

This shows that $x = -1$ is also a valid solution.

Conclusion

In this article, we solved for x in the equation $x = 2 \sqrt{x+1} - 1$. We used algebraic manipulation and properties of square roots to isolate x and find its value. We found two valid solutions: $x = 3$ and $x = -1$. We checked our solutions to make sure they satisfy the original equation, and both solutions were valid.

Final Answer

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Introduction

In our previous article, we solved for x in the equation $x = 2 \sqrt{x+1} - 1$. We used algebraic manipulation and properties of square roots to isolate x and find its value. In this article, we will answer some frequently asked questions about solving for x in this equation.

Q: What is the first step in solving for x in the equation $x = 2 \sqrt{x+1} - 1$?

A: The first step in solving for x in the equation $x = 2 \sqrt{x+1} - 1$ is to isolate the square root term. We can do this by adding 1 to both sides of the equation, which gives us $x + 1 = 2 \sqrt{x+1}$.

Q: Why do we need to square both sides of the equation?

A: We need to square both sides of the equation to eliminate the square root term. When we square both sides, we get $(x + 1)^2 = (2 \sqrt{x+1})^2$, which simplifies to $x^2 + 2x + 1 = 4(x+1)$.

Q: How do we simplify the equation $x^2 + 2x + 1 = 4(x+1)$?

A: We can simplify the equation $x^2 + 2x + 1 = 4(x+1)$ by expanding and simplifying. We get $x^2 + 2x + 1 = 4x + 4$, which simplifies to $x^2 - 2x - 3 = 0$.

Q: How do we solve the quadratic equation $x^2 - 2x - 3 = 0$?

A: We can solve the quadratic equation $x^2 - 2x - 3 = 0$ by factoring. We get $(x - 3)(x + 1) = 0$, which gives us two possible solutions: $x = 3$ and $x = -1$.

Q: How do we check our solutions to make sure they satisfy the original equation?

A: We can check our solutions by plugging them back into the original equation. For example, we can plug $x = 3$ into the original equation to get $3 = 2 \sqrt{3+1} - 1$, which simplifies to $3 = 2 \sqrt{4} - 1$, which further simplifies to $3 = 2(2) - 1$, which finally simplifies to $3 = 4 - 1$, which equals $3 = 3$. This shows that $x = 3$ is a valid solution.

Q: What are some common mistakes to avoid when solving for x in the equation $x = 2 \sqrt{x+1} - 1$?

A: Some common mistakes to avoid when solving for x in the equation $x = 2 \sqrt{x+1} - 1$ include:

• Not isolating the square root term correctly
• Not squaring both sides of the equation correctly
• Not simplifying the equation correctly
• Not checking the solutions correctly

Conclusion

In this article, we answered some frequently asked questions about solving for x in the equation $x = 2 \sqrt{x+1} - 1$. We covered topics such as isolating the square root term, squaring both sides of the equation, simplifying the equation, and checking the solutions. We also discussed some common mistakes to avoid when solving for x in this equation.

Final Answer

The final answer is: $\boxed{3, -1}$