Solve The Equation:${ \frac{9 \sqrt{x} - 7}{3 \sqrt{x}} = \frac{3 \sqrt{x} + 1}{\sqrt{x} + 5} }$

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 $\frac{9 \sqrt{x} - 7}{3 \sqrt{x}} = \frac{3 \sqrt{x} + 1}{\sqrt{x} + 5}$. Our goal is to solve for the variable $x$ and provide a clear, step-by-step explanation of the solution process.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at its structure. We have two fractions on either side of the equation, each containing square roots. The left-hand side of the equation is $\frac{9 \sqrt{x} - 7}{3 \sqrt{x}}$, while the right-hand side is $\frac{3 \sqrt{x} + 1}{\sqrt{x} + 5}$. Our objective is to find the value of $x$ that satisfies this equation.

Step 1: Cross-Multiplication

To solve this equation, we will start by cross-multiplying the two fractions. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. The resulting equation will be:

$$\frac{9 \sqrt{x} - 7}{3 \sqrt{x}} \cdot (\sqrt{x} + 5) = \frac{3 \sqrt{x} + 1}{\sqrt{x} + 5} \cdot 3 \sqrt{x}$$

Step 2: Simplifying the Equation

Now that we have cross-multiplied the fractions, we can simplify the equation by multiplying out the terms. This will give us:

$$(9 \sqrt{x} - 7)(\sqrt{x} + 5) = (3 \sqrt{x} + 1)(3 \sqrt{x})$$

Expanding the left-hand side of the equation, we get:

$$9 \sqrt{x}^2 + 45 \sqrt{x} - 7 \sqrt{x} - 35 = 9 \sqrt{x}^2 + 3 \sqrt{x}$$

Step 3: Combining Like Terms

Next, we can combine like terms on both sides of the equation. On the left-hand side, we have:

$$9 \sqrt{x}^2 + 38 \sqrt{x} - 35$$

On the right-hand side, we have:

$$9 \sqrt{x}^2 + 3 \sqrt{x}$$

Step 4: Subtracting the Right-Hand Side from the Left-Hand Side

Now, we can subtract the right-hand side from the left-hand side to eliminate the $9 \sqrt{x}^2$ term. This gives us:

$$38 \sqrt{x} - 35 = 3 \sqrt{x}$$

Step 5: Isolating the Square Root Term

To isolate the square root term, we can add $35$ to both sides of the equation. This gives us:

$$38 \sqrt{x} = 3 \sqrt{x} + 35$$

Next, we can subtract $3 \sqrt{x}$ from both sides to get:

$$35 \sqrt{x} = 35$$

Step 6: Dividing Both Sides by 35

Finally, we can divide both sides of the equation by $35$ to solve for $\sqrt{x}$. This gives us:

$$\sqrt{x} = 1$$

Step 7: Squaring Both Sides

To find the value of $x$, we can square both sides of the equation. This gives us:

$$x = 1^2$$

Conclusion

In this article, we have solved the equation $\frac{9 \sqrt{x} - 7}{3 \sqrt{x}} = \frac{3 \sqrt{x} + 1}{\sqrt{x} + 5}$ using a step-by-step approach. We started by cross-multiplying the fractions, then simplified the equation by multiplying out the terms. We combined like terms, subtracted the right-hand side from the left-hand side, isolated the square root term, and finally squared both sides to find the value of $x$. The final answer is $x = 1$.

Additional Tips and Tricks

When solving equations involving square roots, it's essential to be careful when multiplying and dividing terms. Make sure to follow the order of operations and simplify the equation step by step. Additionally, be mindful of the signs and exponents when working with square roots.

Real-World Applications

Solving equations involving square roots has numerous real-world applications in fields such as physics, engineering, and computer science. For example, in physics, square roots are used to calculate the velocity and acceleration of objects. In engineering, square roots are used to design and optimize systems, such as bridges and buildings. In computer science, square roots are used in algorithms for image and signal processing.

Common Mistakes to Avoid

When solving equations involving square roots, there are several common mistakes to avoid. These include:

• Not following the order of operations
• Not simplifying the equation step by step
• Not being mindful of the signs and exponents
• Not checking the solution for extraneous solutions

Introduction

In our previous article, we solved the equation $\frac{9 \sqrt{x} - 7}{3 \sqrt{x}} = \frac{3 \sqrt{x} + 1}{\sqrt{x} + 5}$ using a step-by-step approach. In this article, we will answer some of the most frequently asked questions about solving equations involving square roots.

Q: What is the first step in solving an equation involving square roots?

A: The first step in solving an equation involving square roots is to cross-multiply the fractions. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa.

Q: How do I simplify the equation after cross-multiplying?

A: After cross-multiplying, you can simplify the equation by multiplying out the terms. This will give you a new equation with fewer terms. Make sure to follow the order of operations and simplify the equation step by step.

Q: What is the difference between a square root and a square?

A: A square root is the inverse operation of squaring a number. For example, the square root of 16 is 4, because 4 squared is 16. A square, on the other hand, is the result of multiplying a number by itself. For example, the square of 4 is 16.

Q: How do I know if a solution is extraneous?

A: A solution is extraneous if it does not satisfy the original equation. To check if a solution is extraneous, plug it back into the original equation and see if it is true. If it is not true, then the solution is extraneous.

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:

• Not following the order of operations
• Not simplifying the equation step by step
• Not being mindful of the signs and exponents
• Not checking the solution for extraneous solutions

Q: How do I apply the solution to real-world problems?

A: The solution to an equation involving square roots can be applied to real-world problems in a variety of ways. For example, in physics, square roots are used to calculate the velocity and acceleration of objects. In engineering, square roots are used to design and optimize systems, such as bridges and buildings. In computer science, square roots are used in algorithms for image and signal processing.

Q: What are some examples of equations involving square roots?

A: Some examples of equations involving square roots include:

• $\sqrt{x} + 3 = 5$
• $x^2 - 4 = 0$
• $\frac{9 \sqrt{x} - 7}{3 \sqrt{x}} = \frac{3 \sqrt{x} + 1}{\sqrt{x} + 5}$

Q: How do I know if an equation is solvable?

A: An equation is solvable if it has a solution that satisfies the equation. To determine if an equation is solvable, try to solve it using a step-by-step approach. If you are unable to solve the equation, then it may not be solvable.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving equations involving square roots. We have covered topics such as cross-multiplying, simplifying the equation, and checking for extraneous solutions. We have also discussed the importance of following the order of operations and being mindful of the signs and exponents. By following these tips and avoiding common mistakes, you can become proficient in solving equations involving square roots and apply your skills to real-world problems.

Additional Resources

For more information on solving equations involving square roots, check out the following resources:

• Khan Academy: Solving Equations with Square Roots
• Mathway: Solving Equations with Square Roots
• Wolfram Alpha: Solving Equations with Square Roots

Real-World Applications

Solving equations involving square roots has numerous real-world applications in fields such as physics, engineering, and computer science. For example, in physics, square roots are used to calculate the velocity and acceleration of objects. In engineering, square roots are used to design and optimize systems, such as bridges and buildings. In computer science, square roots are used in algorithms for image and signal processing.

Common Mistakes to Avoid

When solving equations involving square roots, there are several common mistakes to avoid. These include:

• Not following the order of operations
• Not simplifying the equation step by step
• Not being mindful of the signs and exponents
• Not checking the solution for extraneous solutions

By following these tips and avoiding common mistakes, you can become proficient in solving equations involving square roots and apply your skills to real-world problems.