Solve The Equation: 9 + X − X = 5 9 + X \sqrt{9+x} - \sqrt{x} = \frac{5}{\sqrt{9+x}} 9 + X ​ − X ​ = 9 + X ​ 5 ​

Introduction

In this article, we will delve into the world of mathematics and explore a complex equation involving square roots. The equation in question is $\sqrt{9+x} - \sqrt{x} = \frac{5}{\sqrt{9+x}}$. Our goal is to solve this equation and provide a clear understanding of the steps involved.

Understanding the Equation

The given equation is a classic example of an equation involving square roots. It consists of two square root terms, $\sqrt{9+x}$ and $\sqrt{x}$, and a fraction $\frac{5}{\sqrt{9+x}}$. To solve this equation, we need to isolate the variable $x$ and simplify the expression.

**Step 1: Multiply Both Sides by $\sqrt{9+x}$$

To eliminate the fraction, we can multiply both sides of the equation by $\sqrt{9+x}$. This will help us to simplify the expression and make it easier to solve.

$\sqrt{9+x} - \sqrt{x} = \frac{5}{\sqrt{9+x}}$

$\sqrt{9+x} \cdot (\sqrt{9+x} - \sqrt{x}) = \frac{5}{\sqrt{9+x}} \cdot \sqrt{9+x}$

$(\sqrt{9+x})^2 - \sqrt{9+x} \cdot \sqrt{x} = 5$

Step 2: Simplify the Expression

Now that we have multiplied both sides by $\sqrt{9+x}$, we can simplify the expression. The left-hand side of the equation can be simplified using the formula $(a-b)^2 = a^2 - 2ab + b^2$.

$(\sqrt{9+x})^2 - \sqrt{9+x} \cdot \sqrt{x} = 5$

$9+x - \sqrt{9+x} \cdot \sqrt{x} = 5$

**Step 3: Isolate the Variable $x$$

Our goal is to isolate the variable $x$ and solve for its value. To do this, we need to get rid of the square root term $\sqrt{9+x} \cdot \sqrt{x}$. We can do this by squaring both sides of the equation.

$9+x - \sqrt{9+x} \cdot \sqrt{x} = 5$

$(9+x - \sqrt{9+x} \cdot \sqrt{x})^2 = 5^2$

Step 4: Expand and Simplify

Now that we have squared both sides of the equation, we can expand and simplify the expression.

$(9+x - \sqrt{9+x} \cdot \sqrt{x})^2 = 5^2$

$81 + 18x + x^2 - 2(9+x)\sqrt{9+x} \cdot \sqrt{x} + (9+x)^2 \cdot x = 25$

Step 5: Simplify and Rearrange

To simplify the expression, we can combine like terms and rearrange the equation.

$81 + 18x + x^2 - 2(9+x)\sqrt{9+x} \cdot \sqrt{x} + (9+x)^2 \cdot x = 25$

$81 + 18x + x^2 - 18\sqrt{9+x} \cdot \sqrt{x} - 2x\sqrt{9+x} \cdot \sqrt{x} + 9x + x^2 + 81x + 9x^2 = 25$

Step 6: Combine Like Terms

Now that we have simplified the expression, we can combine like terms.

$81 + 18x + x^2 - 18\sqrt{9+x} \cdot \sqrt{x} - 2x\sqrt{9+x} \cdot \sqrt{x} + 9x + x^2 + 81x + 9x^2 = 25$

$81 + 108x + 2x^2 - 18\sqrt{9+x} \cdot \sqrt{x} - 2x\sqrt{9+x} \cdot \sqrt{x} = 25$

Step 7: Factor Out the Common Term

To simplify the expression further, we can factor out the common term $\sqrt{9+x} \cdot \sqrt{x}$.

$81 + 108x + 2x^2 - 18\sqrt{9+x} \cdot \sqrt{x} - 2x\sqrt{9+x} \cdot \sqrt{x} = 25$

$81 + 108x + 2x^2 - 2\sqrt{9+x} \cdot \sqrt{x}(9+x) = 25$

Step 8: Simplify and Rearrange

To simplify the expression, we can combine like terms and rearrange the equation.

$81 + 108x + 2x^2 - 2\sqrt{9+x} \cdot \sqrt{x}(9+x) = 25$

$81 + 108x + 2x^2 - 18x - 2x^2 = 25$

Step 9: Combine Like Terms

Now that we have simplified the expression, we can combine like terms.

$81 + 108x + 2x^2 - 18x - 2x^2 = 25$

$81 + 90x = 25$

**Step 10: Solve for $x$$

Our final step is to solve for $x$. We can do this by isolating the variable $x$ and simplifying the expression.

$81 + 90x = 25$

$90x = -56$

$x = -\frac{56}{90}$

$x = -\frac{28}{45}$

Conclusion

$$$$$$$$

Introduction

In our previous article, we solved the equation $\sqrt{9+x} - \sqrt{x} = \frac{5}{\sqrt{9+x}}$ using a step-by-step approach. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is the main concept behind solving this equation?

A: The main concept behind solving this equation is to isolate the variable $x$ and simplify the expression. We used a combination of algebraic manipulations and square root properties to solve for $x$.

Q: Why did we multiply both sides by $\sqrt{9+x}$?

A: We multiplied both sides by $\sqrt{9+x}$ to eliminate the fraction and simplify the expression. This allowed us to work with a simpler equation and make progress towards solving for $x$.

Q: What is the significance of squaring both sides of the equation?

A: Squaring both sides of the equation allowed us to eliminate the square root term and simplify the expression. This was a crucial step in solving for $x$.

Q: How did we simplify the expression after squaring both sides?

A: We simplified the expression by combining like terms and rearranging the equation. This helped us to isolate the variable $x$ and make progress towards solving for its value.

Q: What is the final solution to the equation?

A: The final solution to the equation is $x = -\frac{28}{45}$.

Q: Can you explain the steps involved in solving this equation?

A: Here are the steps involved in solving this equation:

1. Multiply both sides by $\sqrt{9+x}$ to eliminate the fraction.
2. Simplify the expression by combining like terms and rearranging the equation.
3. Square both sides of the equation to eliminate the square root term.
4. Simplify the expression by combining like terms and rearranging the equation.
5. Isolate the variable $x$ and solve for its value.

Q: What are some common mistakes to avoid when solving this equation?

A: Some common mistakes to avoid when solving this equation include:

• Not multiplying both sides by $\sqrt{9+x}$ to eliminate the fraction.
• Not simplifying the expression by combining like terms and rearranging the equation.
• Not squaring both sides of the equation to eliminate the square root term.
• Not isolating the variable $x$ and solving for its value.

Q: Can you provide some tips for solving equations involving square roots?

A: Here are some tips for solving equations involving square roots:

• Always multiply both sides by the square root term to eliminate the fraction.
• Simplify the expression by combining like terms and rearranging the equation.
• Square both sides of the equation to eliminate the square root term.
• Isolate the variable and solve for its value.
• Be careful when simplifying the expression and avoid making common mistakes.

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to the equation $\sqrt{9+x} - \sqrt{x} = \frac{5}{\sqrt{9+x}}$. We covered common questions and provided tips for solving equations involving square roots. We hope this article has been helpful in understanding the solution and answering any questions you may have.