Solve The Equation:$\[ N^3 - N = 7n^2 - 7 \\]

Introduction

In this article, we will delve into the world of cubic equations and explore a specific equation that has been given to us: $n^3 - n = 7n^2 - 7$. Our goal is to solve this equation and find the values of $n$ that satisfy it. We will break down the solution into manageable steps, using algebraic manipulations and factoring techniques to simplify the equation and ultimately find the solutions.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at its structure. The equation is a cubic equation, meaning that the highest power of the variable $n$ is 3. The equation can be written as:

$$n^3 - n = 7n^2 - 7$$

Our objective is to find the values of $n$ that make this equation true.

Step 1: Rearrange the Equation

To begin solving the equation, we can start by rearranging it to get all the terms on one side. This will give us a single expression that we can work with:

$$n^3 - 7n^2 - n + 7 = 0$$

This is a more manageable form of the equation, and we can now start applying algebraic manipulations to simplify it.

Step 2: Factor the Equation

One of the most powerful tools in algebra is factoring. By factoring the equation, we can break it down into simpler expressions that are easier to work with. Let's try factoring the equation:

$$n^3 - 7n^2 - n + 7 = (n^3 - 7n^2) - (n - 7)$$

We can now factor out a common term from each expression:

$$n^2(n - 7) - 1(n - 7)$$

This gives us:

$$(n^2 - 1)(n - 7) = 0$$

Step 3: Solve for $n$

Now that we have factored the equation, we can set each factor equal to zero and solve for $n$. Let's start with the first factor:

$$(n^2 - 1) = 0$$

This gives us:

$$n^2 = 1$$

Taking the square root of both sides, we get:

$$n = \pm 1$$

Now, let's move on to the second factor:

$$(n - 7) = 0$$

This gives us:

$$n = 7$$

Conclusion

In this article, we have solved the cubic equation $n^3 - n = 7n^2 - 7$ using algebraic manipulations and factoring techniques. We have broken down the solution into manageable steps, starting with rearranging the equation and then factoring it. Finally, we have set each factor equal to zero and solved for $n$. The solutions to the equation are $n = -1, 1, 7$.

Final Answer

The final answer is $\boxed{-1, 1, 7}$.

Additional Tips and Tricks

• When solving cubic equations, it's often helpful to start by rearranging the equation to get all the terms on one side.
• Factoring is a powerful tool in algebra, and it can be used to break down complex equations into simpler expressions.
• When setting each factor equal to zero, make sure to consider all possible solutions, including negative and positive values.

Common Mistakes to Avoid

• When rearranging the equation, make sure to get all the terms on one side.
• When factoring, make sure to factor out common terms from each expression.
• When setting each factor equal to zero, make sure to consider all possible solutions, including negative and positive values.

Real-World Applications

Cubic equations have many real-world applications, including:

• Physics: Cubic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Cubic equations are used to design and optimize systems, such as bridges and buildings.
• Computer Science: Cubic equations are used in algorithms and data structures, such as sorting and searching.

Conclusion

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Introduction

In our previous article, we solved the cubic equation $n^3 - n = 7n^2 - 7$ using algebraic manipulations and factoring techniques. In this article, we will provide a Q&A guide to help you understand the solution and apply it to similar problems.

Q: What is a cubic equation?

A: A cubic equation is a polynomial equation of degree 3, meaning that the highest power of the variable is 3. Cubic equations can be written in the form $ax^3 + bx^2 + cx + d = 0$, where $a$, $b$, $c$, and $d$ are constants.

Q: How do I solve a cubic equation?

A: To solve a cubic equation, you can use algebraic manipulations and factoring techniques. Start by rearranging the equation to get all the terms on one side. Then, try to factor the equation by finding common terms and grouping them together. Finally, set each factor equal to zero and solve for the variable.

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

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

• Not getting all the terms on one side of the equation
• Not factoring out common terms
• Not considering all possible solutions, including negative and positive values

Q: How do I factor a cubic equation?

A: To factor a cubic equation, you can try to find common terms and group them together. Look for terms that have a common factor, such as a constant or a variable. Then, factor out the common term and simplify the equation.

Q: What are some real-world applications of cubic equations?

A: Cubic equations have many real-world applications, including:

• Physics: Cubic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Cubic equations are used to design and optimize systems, such as bridges and buildings.
• Computer Science: Cubic equations are used in algorithms and data structures, such as sorting and searching.

Q: How do I check my solution to a cubic equation?

A: To check your solution to a cubic equation, you can plug the value back into the original equation and simplify. If the equation is true, then the value is a solution to the equation.

Q: What are some tips for solving cubic equations?

A: Some tips for solving cubic equations include:

• Start by rearranging the equation to get all the terms on one side.
• Try to factor the equation by finding common terms and grouping them together.
• Set each factor equal to zero and solve for the variable.
• Consider all possible solutions, including negative and positive values.

Q: How do I use technology to solve cubic equations?

A: You can use technology, such as calculators or computer software, to solve cubic equations. Many calculators and computer programs have built-in functions for solving polynomial equations, including cubic equations.

Conclusion

In conclusion, solving cubic equations requires a combination of algebraic manipulations and factoring techniques. By following the steps outlined in this Q&A guide, you can solve cubic equations and apply them to real-world problems. Remember to avoid common mistakes, check your solution, and use technology to simplify the process.

Additional Resources

• Online calculators and computer software for solving polynomial equations
• Algebra textbooks and online resources for learning more about solving cubic equations
• Real-world applications of cubic equations in physics, engineering, and computer science

Final Answer

The final answer is $\boxed{-1, 1, 7}$.