Solve For \[$ X \$\] In The Equation:$\[ \sqrt{4x^2 - 27} = 3 \\]

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Introduction

Mathematics is a vast and fascinating subject that deals with numbers, quantities, and shapes. It is a fundamental tool for problem-solving, critical thinking, and logical reasoning. In mathematics, equations are used to represent relationships between variables, and solving them is an essential skill for anyone interested in mathematics. In this article, we will focus on solving a specific equation involving a square root, and we will explore the steps and techniques required to find the solution.

Understanding the Equation

The given equation is $\sqrt{4x^2 - 27} = 3$. This equation involves a square root, which means that the expression inside the square root must be non-negative. In other words, $4x^2 - 27 \geq 0$. We will start by analyzing the expression inside the square root and then proceed to solve the equation.

Analyzing the Expression Inside the Square Root

The expression inside the square root is $4x^2 - 27$. This is a quadratic expression, which means that it can be factored or solved using the quadratic formula. To analyze this expression, we can start by finding its minimum value. The minimum value of a quadratic expression $ax^2 + bx + c$ is given by $c - \frac{b^2}{4a}$. In this case, $a = 4$, $b = 0$, and $c = -27$. Therefore, the minimum value of the expression is $-27 - \frac{0^2}{4(4)} = -27$.

Finding the Solution

Since the expression inside the square root must be non-negative, we know that $4x^2 - 27 \geq 0$. This means that $4x^2 \geq 27$, which implies that $x^2 \geq \frac{27}{4}$. Taking the square root of both sides, we get $x \geq \frac{3\sqrt{3}}{2}$ or $x \leq -\frac{3\sqrt{3}}{2}$. However, since the square root of a negative number is not defined in the real number system, we can ignore the negative solution.

Solving the Equation

Now that we have found the possible values of $x$, we can proceed to solve the equation. We start by squaring both sides of the equation: $\left(\sqrt{4x^2 - 27}\right)^2 = 3^2$. This simplifies to $4x^2 - 27 = 9$. Adding 27 to both sides, we get $4x^2 = 36$. Dividing both sides by 4, we get $x^2 = 9$. Taking the square root of both sides, we get $x = \pm 3$.

Checking the Solutions

We have found two possible solutions: $x = 3$ and $x = -3$. To check these solutions, we can substitute them back into the original equation. If the equation holds true, then the solution is valid. Substituting $x = 3$ into the original equation, we get $\sqrt{4(3)^2 - 27} = \sqrt{36 - 27} = \sqrt{9} = 3$, which is true. Substituting $x = -3$ into the original equation, we get $\sqrt{4(-3)^2 - 27} = \sqrt{36 - 27} = \sqrt{9} = 3$, which is also true.

Conclusion

In this article, we have solved the equation $\sqrt{4x^2 - 27} = 3$. We started by analyzing the expression inside the square root and found its minimum value. We then found the possible values of $x$ and solved the equation. Finally, we checked the solutions to ensure that they are valid. The solutions to the equation are $x = 3$ and $x = -3$.

Final Answer

The final answer is $\boxed{3}$ and $\boxed{-3}$.

Additional Tips and Tricks

• When solving equations involving square roots, it is essential to analyze the expression inside the square root and ensure that it is non-negative.
• When solving quadratic equations, it is helpful to use the quadratic formula or factor the expression to find the solutions.
• When checking solutions, it is crucial to substitute the values back into the original equation to ensure that they are valid.

Frequently Asked Questions

• Q: What is the minimum value of the expression $4x^2 - 27$? A: The minimum value of the expression is $-27$.
• Q: What are the possible values of $x$? A: The possible values of $x$ are $x \geq \frac{3\sqrt{3}}{2}$ or $x \leq -\frac{3\sqrt{3}}{2}$.
• Q: How do I check the solutions? A: To check the solutions, substitute the values back into the original equation to ensure that they are valid.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for general information purposes only and are not directly related to the specific equation solved in this article.

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Q&A: Frequently Asked Questions

Q: What is the minimum value of the expression $4x^2 - 27$?

A: The minimum value of the expression is $-27$. This is because the expression is a quadratic expression, and the minimum value of a quadratic expression $ax^2 + bx + c$ is given by $c - \frac{b^2}{4a}$. In this case, $a = 4$, $b = 0$, and $c = -27$. Therefore, the minimum value of the expression is $-27 - \frac{0^2}{4(4)} = -27$.

Q: What are the possible values of $x$?

A: The possible values of $x$ are $x \geq \frac{3\sqrt{3}}{2}$ or $x \leq -\frac{3\sqrt{3}}{2}$. This is because the expression inside the square root must be non-negative, which means that $4x^2 - 27 \geq 0$. This implies that $4x^2 \geq 27$, which further implies that $x^2 \geq \frac{27}{4}$. Taking the square root of both sides, we get $x \geq \frac{3\sqrt{3}}{2}$ or $x \leq -\frac{3\sqrt{3}}{2}$.

Q: How do I check the solutions?

A: To check the solutions, substitute the values back into the original equation to ensure that they are valid. For example, if we have a solution $x = 3$, we can substitute this value back into the original equation: $\sqrt{4(3)^2 - 27} = \sqrt{36 - 27} = \sqrt{9} = 3$. If the equation holds true, then the solution is valid.

Q: What is the final answer?

A: The final answer is $\boxed{3}$ and $\boxed{-3}$.

Q: Can I use other methods to solve the equation?

A: Yes, you can use other methods to solve the equation. For example, you can use the quadratic formula or factor the expression to find the solutions.

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 analyzing the expression inside the square root to ensure that it is non-negative.
• Not checking the solutions to ensure that they are valid.
• Not using the correct method to solve the equation.

Q: How do I know if the solution is valid?

A: To know if the solution is valid, you need to substitute the value back into the original equation and check if the equation holds true.

Q: Can I use a calculator to solve the equation?

A: Yes, you can use a calculator to solve the equation. However, it is always a good idea to check the solution by hand to ensure that it is valid.

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 used to calculate distances, velocities, and accelerations.
• Engineering: Solving equations involving square roots is used to design and optimize systems.
• Computer Science: Solving equations involving square roots is used in algorithms and data structures.

Q: Can I use other mathematical operations to solve the equation?

A: Yes, you can use other mathematical operations to solve the equation. For example, you can use addition, subtraction, multiplication, and division to simplify the equation.

Q: What are some common pitfalls to avoid when solving equations involving square roots?

A: Some common pitfalls to avoid when solving equations involving square roots include:

• Not analyzing the expression inside the square root to ensure that it is non-negative.
• Not checking the solutions to ensure that they are valid.
• Not using the correct method to solve the equation.

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

A: To know if the equation is solvable, you need to analyze the expression inside the square root to ensure that it is non-negative.

Q: Can I use a graphing calculator to solve the equation?

A: Yes, you can use a graphing calculator to solve the equation. However, it is always a good idea to check the solution by hand to ensure that it is valid.

Q: What are some real-world examples of solving equations involving square roots?

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

• Calculating distances and velocities in physics.
• Designing and optimizing systems in engineering.
• Developing algorithms and data structures in computer science.

Q: Can I use other mathematical techniques to solve the equation?

A: Yes, you can use other mathematical techniques to solve the equation. For example, you can use the quadratic formula or factor the expression to find the solutions.

Q: What are some common mistakes to avoid when using a calculator to solve the equation?

A: Some common mistakes to avoid when using a calculator to solve the equation include:

• Not checking the solution to ensure that it is valid.
• Not using the correct method to solve the equation.
• Not analyzing the expression inside the square root to ensure that it is non-negative.

Q: How do I know if the calculator is giving me the correct solution?

A: To know if the calculator is giving you the correct solution, you need to check the solution by hand to ensure that it is valid.

Q: Can I use a computer program to solve the equation?

A: Yes, you can use a computer program to solve the equation. However, it is always a good idea to check the solution by hand to ensure that it is valid.

Q: What are some real-world applications of using a computer program to solve the equation?

A: Using a computer program to solve the equation has many real-world applications, including:

• Developing algorithms and data structures in computer science.
• Designing and optimizing systems in engineering.
• Calculating distances and velocities in physics.

Q: Can I use other mathematical software to solve the equation?

A: Yes, you can use other mathematical software to solve the equation. For example, you can use Mathematica or Maple to solve the equation.

Q: What are some common mistakes to avoid when using mathematical software to solve the equation?

A: Some common mistakes to avoid when using mathematical software to solve the equation include:

• Not checking the solution to ensure that it is valid.
• Not using the correct method to solve the equation.
• Not analyzing the expression inside the square root to ensure that it is non-negative.

Q: How do I know if the mathematical software is giving me the correct solution?

A: To know if the mathematical software is giving you the correct solution, you need to check the solution by hand to ensure that it is valid.

Q: Can I use a spreadsheet to solve the equation?

A: Yes, you can use a spreadsheet to solve the equation. However, it is always a good idea to check the solution by hand to ensure that it is valid.

Q: What are some real-world applications of using a spreadsheet to solve the equation?

A: Using a spreadsheet to solve the equation has many real-world applications, including:

• Developing algorithms and data structures in computer science.
• Designing and optimizing systems in engineering.
• Calculating distances and velocities in physics.

Q: Can I use other mathematical tools to solve the equation?

A: Yes, you can use other mathematical tools to solve the equation. For example, you can use a graphing calculator or a computer program to solve the equation.

Q: What are some common mistakes to avoid when using mathematical tools to solve the equation?

A: Some common mistakes to avoid when using mathematical tools to solve the equation include:

• Not checking the solution to ensure that it is valid.
• Not using the correct method to solve the equation.
• Not analyzing the expression inside the square root to ensure that it is non-negative.

Q: How do I know if the mathematical tool is giving me the correct solution?

A: To know if the mathematical tool is giving you the correct solution, you need to check the solution by hand to ensure that it is valid.

Q: Can I use a combination of mathematical tools to solve the equation?

A: Yes, you can use a combination of mathematical tools to solve the equation. For example, you can use a graphing calculator and a computer program to solve the equation.

Q: What are some real-world applications of using a combination of mathematical tools to solve the equation?

A: Using a combination of mathematical tools to solve the equation has many real-world applications, including:

• Developing algorithms and data structures in computer science.
• Designing and optimizing systems in engineering.
• Calculating distances and velocities in physics.

Q: Can I use other mathematical techniques to solve the equation?

A: Yes, you can use other mathematical techniques to solve the equation. For example, you can use the quadratic formula or factor the expression to find the solutions.

Q: What are some common mistakes to avoid when using other mathematical techniques to solve the equation?

A: Some common mistakes to avoid when using other mathematical techniques to solve the equation include:

• Not checking the solution to ensure that it is valid.
• Not using the correct method to solve the equation.
• Not analyzing the expression inside the square root to ensure that it is non-negative.

Q: How do I know if the other mathematical technique is giving me the correct solution?

A: To know if the other mathematical technique is giving you the correct