Solve The Equation:$\sqrt{10-15x}-4=6$

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Introduction

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In this article, we will delve into the world of algebra and focus on solving a specific equation involving a square root. The equation in question is $\sqrt{10-15x}-4=6$. Our goal is to isolate the variable $x$ and find its value. We will break down the solution step by step, using algebraic manipulations and properties of square roots.

Understanding the Equation

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The given equation is $\sqrt{10-15x}-4=6$. To begin solving this equation, we need to understand the properties of square roots. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we have $\sqrt{10-15x}$, which means we are looking for a value that, when squared, gives $10-15x$.

Isolating the Square Root

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Our first step is to isolate the square root term on one side of the equation. We can do this by adding $4$ to both sides of the equation:

$$\sqrt{10-15x}-4+4=6+4$$

This simplifies to:

$$\sqrt{10-15x}=10$$

Squaring Both Sides

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Next, we will square both sides of the equation to eliminate the square root. Squaring both sides gives us:

$$(\sqrt{10-15x})^2=(10)^2$$

Using the property of square roots, we can simplify this to:

$$10-15x=100$$

Solving for $x$

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Now that we have a linear equation, we can solve for $x$. We will start by subtracting $10$ from both sides of the equation:

$$-15x=90$$

Next, we will divide both sides of the equation by $-15$ to isolate $x$:

$$x=-\frac{90}{15}$$

Simplifying this expression gives us:

$$x=-6$$

Checking the Solution

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To ensure that our solution is correct, we will substitute $x=-6$ back into the original equation and check if it is true:

$$\sqrt{10-15(-6)}-4=6$$

Simplifying the expression inside the square root gives us:

$$\sqrt{10+90}-4=6$$

This simplifies to:

$$\sqrt{100}-4=6$$

Which is equivalent to:

$$10-4=6$$

This is indeed true, so we can confirm that our solution is correct.

Conclusion

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In this article, we solved the equation $\sqrt{10-15x}-4=6$ by isolating the square root term, squaring both sides, and solving for $x$. We found that the value of $x$ is $-6$. We also checked our solution by substituting $x=-6$ back into the original equation and confirmed that it is true. This demonstrates the importance of checking our solutions to ensure that they are correct.

Final Thoughts

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Solving equations involving square roots can be challenging, but with the right approach and techniques, we can find the solution. In this case, we used algebraic manipulations and properties of square roots to isolate the variable $x$ and find its value. We hope that this article has provided a clear and concise explanation of how to solve equations involving square roots.

Additional Resources

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If you are struggling with solving equations involving square roots, here are some additional resources that may help:

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

These resources provide step-by-step instructions and examples to help you understand how to solve equations involving square roots.

Common Mistakes

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

• Not isolating the square root term
• Not squaring both sides of the equation
• Not checking the solution

By avoiding these common mistakes, you can ensure that your solutions are correct and accurate.

Real-World Applications

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Solving equations involving square roots has many real-world applications, including:

• Physics: Solving equations involving square roots is essential in physics, particularly in the study of motion and energy.
• Engineering: Solving equations involving square roots is critical in engineering, particularly in the design of structures and systems.
• Computer Science: Solving equations involving square roots is essential in computer science, particularly in the development of algorithms and data structures.

By understanding how to solve equations involving square roots, you can apply this knowledge to a wide range of real-world problems and challenges.

Conclusion

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In conclusion, solving the equation $\sqrt{10-15x}-4=6$ requires a clear understanding of algebraic manipulations and properties of square roots. By isolating the square root term, squaring both sides, and solving for $x$, we can find the value of $x$. We hope that this article has provided a clear and concise explanation of how to solve equations involving square roots.

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Introduction

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In our previous article, we solved the equation $\sqrt{10-15x}-4=6$ by isolating the square root term, squaring both sides, and solving for $x$. We found that the value of $x$ is $-6$. In this article, we will answer some frequently asked questions related to solving equations involving square roots.

Q&A

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

A: The first step in solving an equation involving a square root is to isolate the square root term on one side of the equation.

Q: How do I isolate the square root term?

A: To isolate the square root term, you can add or subtract the same value from both sides of the equation. For example, if the equation is $\sqrt{10-15x}-4=6$, you can add $4$ to both sides to get $\sqrt{10-15x}=10$.

Q: What is the next step after isolating the square root term?

A: After isolating the square root term, the next step is to square both sides of the equation. This will eliminate the square root and allow you to solve for the variable.

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

A: You need to square both sides of the equation to eliminate the square root. This is because the square root of a number is a value that, when multiplied by itself, gives the original number. By squaring both sides, you are essentially multiplying both sides by themselves, which eliminates the square root.

Q: How do I solve for the variable after squaring both sides?

A: After squaring both sides, you can solve for the variable by simplifying the equation and isolating the variable. For example, if the equation is $10-15x=100$, you can subtract $10$ from both sides to get $-15x=90$, and then divide both sides by $-15$ to get $x=-6$.

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 isolating the square root term
• Not squaring both sides of the equation
• Not checking the solution

Q: How do I check my solution?

A: To check your solution, you can substitute the value of the variable back into the original equation and see if it is true. For example, if the original equation is $\sqrt{10-15x}-4=6$ and you found that $x=-6$, you can substitute $x=-6$ back into the equation and see if it is true.

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 essential in physics, particularly in the study of motion and energy.
• Engineering: Solving equations involving square roots is critical in engineering, particularly in the design of structures and systems.
• Computer Science: Solving equations involving square roots is essential in computer science, particularly in the development of algorithms and data structures.

Conclusion

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In conclusion, solving equations involving square roots requires a clear understanding of algebraic manipulations and properties of square roots. By isolating the square root term, squaring both sides, and solving for the variable, we can find the value of the variable. We hope that this article has provided a clear and concise explanation of how to solve equations involving square roots and has answered some frequently asked questions.

Additional Resources

---

If you are struggling with solving equations involving square roots, here are some additional resources that may help:

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

These resources provide step-by-step instructions and examples to help you understand how to solve equations involving square roots.

Common Mistakes

---

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

• Not isolating the square root term
• Not squaring both sides of the equation
• Not checking the solution

By avoiding these common mistakes, you can ensure that your solutions are correct and accurate.

Real-World Applications

---

Solving equations involving square roots has many real-world applications, including:

• Physics: Solving equations involving square roots is essential in physics, particularly in the study of motion and energy.
• Engineering: Solving equations involving square roots is critical in engineering, particularly in the design of structures and systems.
• Computer Science: Solving equations involving square roots is essential in computer science, particularly in the development of algorithms and data structures.

By understanding how to solve equations involving square roots, you can apply this knowledge to a wide range of real-world problems and challenges.