Solve For { X $}$ In The Equation:${ \sqrt{\frac{a+x}{a-x}} = 2 }$

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Introduction

In this article, we will delve into solving for x in the given equation: $\sqrt{\frac{a+x}{a-x}} = 2$. This equation involves a square root and a fraction, making it a bit more complex than a standard linear equation. We will break down the solution step by step, using algebraic manipulations to isolate the variable x.

Understanding the Equation

The given equation is $\sqrt{\frac{a+x}{a-x}} = 2$. To start solving for x, we need to understand the properties of square roots and fractions. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, the square root of $\frac{a+x}{a-x}$ is equal to 2.

Step 1: Square Both Sides

To eliminate the square root, we can square both sides of the equation. This will give us:

$$\left(\sqrt{\frac{a+x}{a-x}}\right)^2 = 2^2$$

Simplifying the left-hand side, we get:

$$\frac{a+x}{a-x} = 4$$

Step 2: Multiply Both Sides by (a-x)

To get rid of the fraction, we can multiply both sides of the equation by (a-x). This will give us:

$$\frac{a+x}{a-x} \cdot (a-x) = 4 \cdot (a-x)$$

Simplifying the left-hand side, we get:

$$a+x = 4(a-x)$$

Step 3: Distribute the 4

To simplify the right-hand side, we can distribute the 4:

$$a+x = 4a - 4x$$

Step 4: Add 4x to Both Sides

To isolate the variable x, we can add 4x to both sides of the equation:

$$a+x+4x = 4a - 4x + 4x$$

Simplifying the left-hand side, we get:

$$a+5x = 4a$$

Step 5: Subtract a from Both Sides

To further isolate the variable x, we can subtract a from both sides of the equation:

$$a+5x-a = 4a-a$$

Simplifying the left-hand side, we get:

$$5x = 3a$$

Step 6: Divide Both Sides by 5

To solve for x, we can divide both sides of the equation by 5:

$$\frac{5x}{5} = \frac{3a}{5}$$

Simplifying the left-hand side, we get:

$$x = \frac{3a}{5}$$

Conclusion

In this article, we solved for x in the equation $\sqrt{\frac{a+x}{a-x}} = 2$. We used algebraic manipulations to isolate the variable x, including squaring both sides, multiplying both sides by (a-x), distributing the 4, adding 4x to both sides, subtracting a from both sides, and dividing both sides by 5. The final solution is $x = \frac{3a}{5}$.

Example Use Case

Suppose we have the equation $\sqrt{\frac{2+x}{2-x}} = 2$. We can use the solution we derived earlier to solve for x. Plugging in a = 2, we get:

$$x = \frac{3(2)}{5}$$

Simplifying the right-hand side, we get:

$$x = \frac{6}{5}$$

Therefore, the solution to the equation $\sqrt{\frac{2+x}{2-x}} = 2$ is $x = \frac{6}{5}$.

Tips and Variations

• To solve for x in the equation $\sqrt{\frac{a-x}{a+x}} = 2$, we can use a similar approach, but with a few modifications. We can start by squaring both sides, then multiplying both sides by (a+x), and so on.
• To solve for x in the equation $\sqrt{\frac{a+x}{a-x}} = -2$, we can use a similar approach, but with a few modifications. We can start by squaring both sides, then multiplying both sides by (a-x), and so on.
• To solve for x in the equation $\sqrt{\frac{a-x}{a+x}} = -2$, we can use a similar approach, but with a few modifications. We can start by squaring both sides, then multiplying both sides by (a+x), and so on.

Common Mistakes

• One common mistake when solving for x in the equation $\sqrt{\frac{a+x}{a-x}} = 2$ is to forget to square both sides. This can lead to an incorrect solution.
• Another common mistake is to multiply both sides by (a-x) without checking if (a-x) is equal to zero. This can lead to an incorrect solution.
• A third common mistake is to forget to add 4x to both sides, or to subtract a from both sides. This can lead to an incorrect solution.

Conclusion

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Introduction

In our previous article, we solved for x in the equation $\sqrt{\frac{a+x}{a-x}} = 2$. We used algebraic manipulations to isolate the variable x, including squaring both sides, multiplying both sides by (a-x), distributing the 4, adding 4x to both sides, subtracting a from both sides, and dividing both sides by 5. In this article, we will answer some common questions related to solving for x in this equation.

Q: What is the main concept behind solving for x in the equation $\sqrt{\frac{a+x}{a-x}} = 2$?

A: The main concept behind solving for x in the equation $\sqrt{\frac{a+x}{a-x}} = 2$ is to isolate the variable x using algebraic manipulations. We start by squaring both sides of the equation, then multiply both sides by (a-x), distribute the 4, add 4x to both sides, subtract a from both sides, and finally divide both sides by 5.

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. Squaring both sides of the equation allows us to get rid of the square root and work with a simpler equation.

Q: What happens if we multiply both sides by (a-x) without checking if (a-x) is equal to zero?

A: If we multiply both sides by (a-x) without checking if (a-x) is equal to zero, we may end up with an incorrect solution. This is because (a-x) is a factor in the equation, and if it is equal to zero, the equation becomes undefined.

Q: Can we use the same approach to solve for x in the equation $\sqrt{\frac{a-x}{a+x}} = 2$?

A: Yes, we can use a similar approach to solve for x in the equation $\sqrt{\frac{a-x}{a+x}} = 2$. However, we need to modify the steps slightly. We can start by squaring both sides, then multiplying both sides by (a+x), and so on.

Q: What is the final solution to the equation $\sqrt{\frac{a+x}{a-x}} = 2$?

A: The final solution to the equation $\sqrt{\frac{a+x}{a-x}} = 2$ is $x = \frac{3a}{5}$.

Q: Can we use the same approach to solve for x in the equation $\sqrt{\frac{a+x}{a-x}} = -2$?

A: Yes, we can use a similar approach to solve for x in the equation $\sqrt{\frac{a+x}{a-x}} = -2$. However, we need to modify the steps slightly. We can start by squaring both sides, then multiplying both sides by (a-x), and so on.

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

A: Some common mistakes to avoid when solving for x in the equation $\sqrt{\frac{a+x}{a-x}} = 2$ include forgetting to square both sides, multiplying both sides by (a-x) without checking if (a-x) is equal to zero, and forgetting to add 4x to both sides or subtract a from both sides.

Conclusion

In conclusion, solving for x in the equation $\sqrt{\frac{a+x}{a-x}} = 2$ requires careful algebraic manipulations. We used squaring both sides, multiplying both sides by (a-x), distributing the 4, adding 4x to both sides, subtracting a from both sides, and dividing both sides by 5 to isolate the variable x. We also answered some common questions related to solving for x in this equation.