What Are The Solutions Of The Equation X 3 − 2 X 2 − 3 X = − 6 X^3 - 2x^2 - 3x = -6 X 3 − 2 X 2 − 3 X = − 6 ?A. X = − 6 , 2 , 3 X = -6, 2, 3 X = − 6 , 2 , 3 B. X = − 3 , − 2 , 3 X = -3, -2, 3 X = − 3 , − 2 , 3 C. X = 2 , 3 , 6 X = 2, 3, 6 X = 2 , 3 , 6 D. X = − 3 , 3 , 2 X = -\sqrt{3}, \sqrt{3}, 2 X = − 3 ​ , 3 ​ , 2 E. X = − 2 , − 3 , 3 X = -2, -\sqrt{3}, \sqrt{3} X = − 2 , − 3 ​ , 3 ​

Introduction

Solving cubic equations can be a challenging task, especially when they are not in the form of a perfect cube. In this article, we will explore the solutions of the equation $x^3 - 2x^2 - 3x = -6$. We will use algebraic methods to find the solutions and compare them with the given options.

Understanding the Equation

The given equation is a cubic equation in the form of $ax^3 + bx^2 + cx + d = 0$. In this case, the equation is $x^3 - 2x^2 - 3x = -6$, which can be rewritten as $x^3 - 2x^2 - 3x + 6 = 0$. To solve this equation, we need to find the values of $x$ that satisfy the equation.

Rearranging the Equation

To make the equation easier to solve, we can rearrange it by moving all the terms to one side of the equation. This gives us $x^3 - 2x^2 - 3x + 6 = 0$. Now, we can try to factorize the equation or use other algebraic methods to find the solutions.

Factoring the Equation

One way to solve the equation is to try to factorize it. We can start by looking for common factors among the terms. In this case, we can factor out a $-3$ from the first three terms, which gives us $-3(x^2 - 2x - 2) + 6 = 0$. Now, we can try to factorize the quadratic expression inside the parentheses.

Factoring the Quadratic Expression

The quadratic expression inside the parentheses is $x^2 - 2x - 2$. We can try to factorize this expression by finding two numbers whose product is $-2$ and whose sum is $-2$. These numbers are $-2$ and $1$, so we can write the quadratic expression as $(x - 2)(x + 1)$. Now, we can substitute this factorization back into the original equation.

Substituting the Factorization

Substituting the factorization back into the original equation gives us $-3(x - 2)(x + 1) + 6 = 0$. Now, we can try to solve for $x$ by setting each factor equal to zero.

Setting Each Factor Equal to Zero

Setting each factor equal to zero gives us three possible solutions: $x - 2 = 0$, $x + 1 = 0$, and $-3 = 0$. The first two solutions are $x = 2$ and $x = -1$, but the third solution is not possible since $-3$ cannot be equal to zero.

Checking the Solutions

To check the solutions, we can substitute each value back into the original equation. If the equation is true, then the value is a solution. Let's start with $x = 2$. Substituting $x = 2$ into the original equation gives us $2^3 - 2(2)^2 - 3(2) = -6$, which simplifies to $8 - 8 - 6 = -6$. This is true, so $x = 2$ is a solution.

Checking the Second Solution

Next, let's check the second solution, $x = -1$. Substituting $x = -1$ into the original equation gives us $(-1)^3 - 2(-1)^2 - 3(-1) = -6$, which simplifies to $-1 - 2 + 3 = 0$. This is not true, so $x = -1$ is not a solution.

Checking the Third Solution

Finally, let's check the third solution, $x = -\sqrt{3}$. Substituting $x = -\sqrt{3}$ into the original equation gives us $(-\sqrt{3})^3 - 2(-\sqrt{3})^2 - 3(-\sqrt{3}) = -6$, which simplifies to $-3\sqrt{3} - 6 + 3\sqrt{3} = -6$. This is true, so $x = -\sqrt{3}$ is a solution.

Checking the Fourth Solution

Next, let's check the fourth solution, $x = \sqrt{3}$. Substituting $x = \sqrt{3}$ into the original equation gives us $(\sqrt{3})^3 - 2(\sqrt{3})^2 - 3(\sqrt{3}) = -6$, which simplifies to $3\sqrt{3} - 6 - 3\sqrt{3} = -6$. This is true, so $x = \sqrt{3}$ is a solution.

Conclusion

In conclusion, the solutions of the equation $x^3 - 2x^2 - 3x = -6$ are $x = 2$, $x = -\sqrt{3}$, and $x = \sqrt{3}$. These solutions match option D, which is $x = -\sqrt{3}, \sqrt{3}, 2$. Therefore, the correct answer is option D.

Final Answer

The final answer is option D, which is $x = -\sqrt{3}, \sqrt{3}, 2$.

Introduction

In our previous article, we explored the solutions of the equation $x^3 - 2x^2 - 3x = -6$. We used algebraic methods to find the solutions and compared them with the given options. In this article, we will answer some frequently asked questions about the solutions of the equation.

Q: What is the equation $x^3 - 2x^2 - 3x = -6$?

A: The equation $x^3 - 2x^2 - 3x = -6$ is a cubic equation in the form of $ax^3 + bx^2 + cx + d = 0$. In this case, the equation is $x^3 - 2x^2 - 3x + 6 = 0$.

Q: How do I solve the equation $x^3 - 2x^2 - 3x = -6$?

A: To solve the equation, you can try to factorize it or use other algebraic methods. One way to solve the equation is to try to factor out a $-3$ from the first three terms, which gives us $-3(x^2 - 2x - 2) + 6 = 0$. Now, you can try to factorize the quadratic expression inside the parentheses.

Q: What are the solutions of the equation $x^3 - 2x^2 - 3x = -6$?

A: The solutions of the equation $x^3 - 2x^2 - 3x = -6$ are $x = 2$, $x = -\sqrt{3}$, and $x = \sqrt{3}$.

Q: How do I check the solutions of the equation $x^3 - 2x^2 - 3x = -6$?

A: To check the solutions, you can substitute each value back into the original equation. If the equation is true, then the value is a solution.

Q: What is the final answer to the equation $x^3 - 2x^2 - 3x = -6$?

A: The final answer to the equation $x^3 - 2x^2 - 3x = -6$ is option D, which is $x = -\sqrt{3}, \sqrt{3}, 2$.

Q: Can I use other methods to solve the equation $x^3 - 2x^2 - 3x = -6$?

A: Yes, you can use other methods to solve the equation $x^3 - 2x^2 - 3x = -6$. Some other methods include using the rational root theorem, synthetic division, or numerical methods.

Q: What are some common mistakes to avoid when solving the equation $x^3 - 2x^2 - 3x = -6$?

A: Some common mistakes to avoid when solving the equation $x^3 - 2x^2 - 3x = -6$ include:

• Not checking the solutions carefully
• Not using the correct method to solve the equation
• Not factoring the equation correctly
• Not using the correct values for the solutions

Conclusion

In conclusion, the solutions of the equation $x^3 - 2x^2 - 3x = -6$ are $x = 2$, $x = -\sqrt{3}$, and $x = \sqrt{3}$. We hope that this Q&A article has been helpful in answering some of the frequently asked questions about the solutions of the equation.

Final Answer

The final answer is option D, which is $x = -\sqrt{3}, \sqrt{3}, 2$.