Solve For \[$x\$\].$\[ 3x^2 = 27 \\]

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it's essential to understand how to approach these types of problems. In this article, we'll focus on solving the quadratic equation $3x^2 = 27$ to find the value of $x$. We'll break down the steps involved in solving this equation and provide a clear explanation of the process.

Understanding the Equation

The given equation is $3x^2 = 27$. This is a quadratic equation in the form of $ax^2 = b$, where $a = 3$ and $b = 27$. Our goal is to isolate the variable $x$ and find its value.

Step 1: Divide Both Sides by 3

To start solving the equation, we need to isolate the term with the variable $x$. We can do this by dividing both sides of the equation by 3. This will give us:

$$x^2 = \frac{27}{3}$$

Step 2: Simplify the Right-Hand Side

Now, we can simplify the right-hand side of the equation by dividing 27 by 3. This gives us:

$$x^2 = 9$$

Step 3: Take the Square Root of Both Sides

To find the value of $x$, we need to take the square root of both sides of the equation. This will give us:

$$x = \pm \sqrt{9}$$

Step 4: Simplify the Square Root

The square root of 9 is 3, so we can simplify the equation to:

$$x = \pm 3$$

Conclusion

In this article, we solved the quadratic equation $3x^2 = 27$ to find the value of $x$. We broke down the steps involved in solving this equation and provided a clear explanation of the process. By following these steps, we were able to isolate the variable $x$ and find its value.

Final Answer

The final answer to the equation $3x^2 = 27$ is $x = \pm 3$.

Related Topics

• Solving quadratic equations
• Quadratic formula
• Square roots
• Algebraic manipulations

Example Problems

• Solve the equation $2x^2 = 16$
• Solve the equation $x^2 = 25$
• Solve the equation $4x^2 = 36$

Tips and Tricks

• When solving quadratic equations, it's essential to isolate the variable $x$.
• Use algebraic manipulations to simplify the equation.
• Take the square root of both sides of the equation to find the value of $x$.
• Be careful when simplifying the square root, as it may have both positive and negative values.

Common Mistakes

• Failing to isolate the variable $x$.
• Not simplifying the equation properly.
• Not taking the square root of both sides of the equation.
• Not being careful when simplifying the square root.

Real-World Applications

• Solving quadratic equations has numerous real-world applications, including physics, engineering, and economics.
• Quadratic equations are used to model real-world phenomena, such as the motion of objects and the growth of populations.
• Algebraic manipulations are essential in solving quadratic equations and have numerous applications in various fields.

Conclusion

Solving quadratic equations is a fundamental concept in mathematics, and it's essential to understand how to approach these types of problems. By following the steps outlined in this article, we can solve quadratic equations and find the value of $x$. We'll continue to explore more complex topics in mathematics and provide a clear explanation of the concepts involved.

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Introduction

In our previous article, we solved the quadratic equation $3x^2 = 27$ to find the value of $x$. We broke down the steps involved in solving this equation and provided a clear explanation of the process. In this article, we'll answer some frequently asked questions related to solving quadratic equations.

Q&A

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It's in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to isolate the variable $x$. You can do this by using algebraic manipulations, such as adding or subtracting the same value to both sides of the equation, or multiplying or dividing both sides by the same value.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It's in the form of $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression and solve for $x$.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. The quadratic formula is a formula that can be used to solve quadratic equations, while factoring involves expressing the quadratic equation as a product of two binomials.

Q: When should I use the quadratic formula and when should I use factoring?

A: You should use the quadratic formula when the quadratic equation cannot be factored easily, or when the equation is in the form of $ax^2 + bx + c = 0$. You should use factoring when the quadratic equation can be expressed as a product of two binomials.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Failing to isolate the variable $x$
• Not simplifying the equation properly
• Not taking the square root of both sides of the equation
• Not being careful when simplifying the square root

Q: How do I check my answer when solving a quadratic equation?

A: To check your answer when solving a quadratic equation, you can plug the value of $x$ back into the original equation and simplify. If the equation is true, then your answer is correct.

Conclusion

Solving quadratic equations is a fundamental concept in mathematics, and it's essential to understand how to approach these types of problems. By following the steps outlined in this article, we can solve quadratic equations and find the value of $x$. We'll continue to explore more complex topics in mathematics and provide a clear explanation of the concepts involved.

Final Answer

The final answer to the equation $3x^2 = 27$ is $x = \pm 3$.

Related Topics

• Solving quadratic equations
• Quadratic formula
• Square roots
• Algebraic manipulations

Example Problems

• Solve the equation $2x^2 = 16$
• Solve the equation $x^2 = 25$
• Solve the equation $4x^2 = 36$

Tips and Tricks

• When solving quadratic equations, it's essential to isolate the variable $x$.
• Use algebraic manipulations to simplify the equation.
• Take the square root of both sides of the equation to find the value of $x$.
• Be careful when simplifying the square root, as it may have both positive and negative values.

Common Mistakes

• Failing to isolate the variable $x$.
• Not simplifying the equation properly.
• Not taking the square root of both sides of the equation.
• Not being careful when simplifying the square root.

Real-World Applications

• Solving quadratic equations has numerous real-world applications, including physics, engineering, and economics.
• Quadratic equations are used to model real-world phenomena, such as the motion of objects and the growth of populations.
• Algebraic manipulations are essential in solving quadratic equations and have numerous applications in various fields.