Solve The Equation:$\sqrt{7x + 3} - \sqrt{10x - 18} = 0$

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Introduction

Solving equations involving square roots can be a challenging task in mathematics. These types of equations often require a combination of algebraic manipulations and careful analysis to isolate the variable. In this article, we will focus on solving the equation $\sqrt{7x + 3} - \sqrt{10x - 18} = 0$. This equation involves two square roots, and our goal is to find the value of $x$ that satisfies this equation.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its structure. The equation consists of two square roots, $\sqrt{7x + 3}$ and $\sqrt{10x - 18}$, which are being subtracted from each other. The result of this subtraction is equal to zero. This means that the two square roots must be equal to each other, as the difference between them is zero.

Setting Up the Equation

To solve the equation, we can start by setting the two square roots equal to each other. This gives us the equation:

$$\sqrt{7x + 3} = \sqrt{10x - 18}$$

Squaring Both Sides

One common technique for solving equations involving square roots is to square both sides of the equation. This will eliminate the square roots and allow us to work with a simpler equation. Squaring both sides of the equation above gives us:

$$7x + 3 = 10x - 18$$

Simplifying the Equation

Now that we have squared both sides of the equation, we can simplify it by combining like terms. Subtracting $7x$ from both sides of the equation gives us:

$$3 = 3x - 18$$

Adding $18$ to both sides of the equation gives us:

$$21 = 3x$$

Solving for $x$

Now that we have simplified the equation, we can solve for $x$. Dividing both sides of the equation by $3$ gives us:

$$x = \frac{21}{3}$$

Evaluating the Solution

The solution to the equation is $x = 7$. To verify this solution, we can plug it back into the original equation and check if it satisfies the equation. Substituting $x = 7$ into the original equation gives us:

$$\sqrt{7(7) + 3} - \sqrt{10(7) - 18} = \sqrt{49 + 3} - \sqrt{70 - 18} = \sqrt{52} - \sqrt{52} = 0$$

Conclusion

In this article, we have solved the equation $\sqrt{7x + 3} - \sqrt{10x - 18} = 0$. We started by setting up the equation and then squared both sides to eliminate the square roots. We then simplified the equation and solved for $x$. The solution to the equation is $x = 7$, which we verified by plugging it back into the original equation.

Additional Tips and Tricks

When solving equations involving square roots, it's essential to be careful with the order of operations. Make sure to square both sides of the equation and simplify it carefully to avoid making mistakes. Additionally, be sure to check your solution by plugging it back into the original equation to verify that it satisfies the equation.

Common Mistakes to Avoid

When solving equations involving square roots, there are several common mistakes to avoid. One common mistake is to forget to square both sides of the equation, which can lead to incorrect solutions. Another common mistake is to simplify the equation incorrectly, which can also lead to incorrect solutions. Be sure to double-check your work and verify your solution by plugging it back into the original equation.

Real-World Applications

Solving equations involving square roots has many real-world applications. For example, in physics, square roots are used to calculate the speed of an object given its kinetic energy. In engineering, square roots are used to calculate the stress on a material given its cross-sectional area. In finance, square roots are used to calculate the volatility of a stock given its historical price data.

Final Thoughts

Solving equations involving square roots can be a challenging task, but with practice and patience, it can be mastered. By following the steps outlined in this article and being careful with the order of operations, you can solve even the most complex equations involving square roots. Remember to always verify your solution by plugging it back into the original equation to ensure that it satisfies the equation.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Engineers" by John Bird

Further Reading

• [1] "Solving Equations Involving Square Roots" by Math Open Reference
• [2] "Square Root Equations" by Purplemath
• [3] "Solving Equations with Square Roots" by Khan Academy
  

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Introduction

In our previous article, we solved the equation $\sqrt{7x + 3} - \sqrt{10x - 18} = 0$. We received many questions from readers who were struggling to understand the solution. In this article, we will address some of the most common questions and provide additional clarification on the solution.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to set the two square roots equal to each other. This gives us the equation $\sqrt{7x + 3} = \sqrt{10x - 18}$.

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 roots. Squaring both sides of the equation gives us $7x + 3 = 10x - 18$.

Q: How do we simplify the equation?

A: We simplify the equation by combining like terms. Subtracting $7x$ from both sides of the equation gives us $3 = 3x - 18$. Adding $18$ to both sides of the equation gives us $21 = 3x$.

Q: How do we solve for $x$?

A: We solve for $x$ by dividing both sides of the equation by $3$. This gives us $x = \frac{21}{3}$.

Q: What is the solution to the equation?

A: The solution to the equation is $x = 7$.

Q: How do we verify the solution?

A: We verify the solution by plugging it back into the original equation. Substituting $x = 7$ into the original equation gives us $\sqrt{7(7) + 3} - \sqrt{10(7) - 18} = \sqrt{49 + 3} - \sqrt{70 - 18} = \sqrt{52} - \sqrt{52} = 0$.

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 forgetting to square both sides of the equation, simplifying the equation incorrectly, and not verifying the solution.

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 calculating the speed of an object given its kinetic energy, calculating the stress on a material given its cross-sectional area, and calculating the volatility of a stock given its historical price data.

Q: What are some additional resources for learning more about solving equations involving square roots?

A: Some additional resources for learning more about solving equations involving square roots include the book "Algebra and Trigonometry" by Michael Sullivan, the book "Calculus" by Michael Spivak, and the website Math Open Reference.

Q: How can I practice solving equations involving square roots?

A: You can practice solving equations involving square roots by working through examples and exercises in a textbook or online resource. You can also try solving equations involving square roots on your own and then checking your solutions with a calculator or online tool.

Q: What are some tips for mastering the skill of solving equations involving square roots?

A: Some tips for mastering the skill of solving equations involving square roots include practicing regularly, paying close attention to the order of operations, and verifying your solutions carefully.

Conclusion

Solving equations involving square roots can be a challenging task, but with practice and patience, it can be mastered. By following the steps outlined in this article and being careful with the order of operations, you can solve even the most complex equations involving square roots. Remember to always verify your solution by plugging it back into the original equation to ensure that it satisfies the equation.

Additional Resources

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Engineers" by John Bird
• [4] "Solving Equations Involving Square Roots" by Math Open Reference
• [5] "Square Root Equations" by Purplemath
• [6] "Solving Equations with Square Roots" by Khan Academy