For What Values Of $x$ Is $x^3 + 27x^2 + 206x + 480 = 0$ A True Statement?List The Values Separated By Commas And Do Not Include $ X = X= X = [/tex] In Your Answer. Provide Your Answer Below:

Introduction

In mathematics, solving cubic equations is a crucial aspect of algebraic manipulation. A cubic equation is a polynomial equation of degree three, which means the highest power of the variable is three. In this article, we will focus on solving the cubic equation ${x^3 + 27x^2 + 206x + 480 = 0}$ to find the values of ${x}$ that make the equation true.

Understanding the Equation

The given equation is a cubic equation in the form of ${ax^3 + bx^2 + cx + d = 0}$, where ${a = 1}$, ${b = 27}$, ${c = 206}$, and ${d = 480}$. To solve this equation, we need to find the values of ${x}$ that satisfy the equation.

Factoring the Equation

One way to solve a cubic equation is to factor it. Factoring involves expressing the equation as a product of simpler equations. In this case, we can try to factor the equation by grouping terms.

x^3 + 27x^2 + 206x + 480 = (x^3 + 27x^2) + (206x + 480)

Grouping Terms

We can group the terms in the equation as follows:

x^3 + 27x^2 + 206x + 480 = x^2(x + 27) + 206x + 480

Factoring by Grouping

Now, we can factor the equation by grouping the terms:

x^2(x + 27) + 206x + 480 = (x^2 + 206)(x + 27) + 480

Simplifying the Equation

We can simplify the equation by combining like terms:

(x^2 + 206)(x + 27) + 480 = (x^2 + 206x + 5613) + 480

Solving the Equation

Now, we can solve the equation by setting it equal to zero:

(x^2 + 206x + 5613) + 480 = 0

Rearranging the Equation

We can rearrange the equation to get:

x^2 + 206x + 6093 = 0

Solving the Quadratic Equation

We can solve the quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Plugging in the Values

We can plug in the values of ${a}$, ${b}$, and ${c}$ into the quadratic formula:

x = (-(206) ± √((206)^2 - 4(1)(6093))) / 2(1)

Simplifying the Expression

We can simplify the expression under the square root:

x = (-206 ± √(42536 - 24372)) / 2

Simplifying the Expression

We can simplify the expression under the square root:

x = (-206 ± √18164) / 2

Simplifying the Expression

We can simplify the expression under the square root:

x = (-206 ± 134) / 2

Solving for x

We can solve for ${x}$ by considering both the positive and negative cases:

x = (-206 + 134) / 2 or x = (-206 - 134) / 2

Simplifying the Expression

We can simplify the expression:

x = -72 / 2 or x = -340 / 2

Simplifying the Expression

We can simplify the expression:

x = -36 or x = -170

Conclusion

In conclusion, the values of ${x}$ that satisfy the equation ${x^3 + 27x^2 + 206x + 480 = 0}$ are ${-36}$ and ${-170}$.

Final Answer

The final answer is: -36, -170

Introduction

In our previous article, we solved the cubic equation ${x^3 + 27x^2 + 206x + 480 = 0}$ to find the values of ${x}$ that make the equation true. In this article, we will answer some frequently asked questions related to solving cubic equations.

Q: What is a cubic equation?

A: A cubic equation is a polynomial equation of degree three, which means the highest power of the variable is three. It is typically written in the form of ${ax^3 + bx^2 + cx + d = 0}$, where ${a}$, ${b}$, ${c}$, and ${d}$ are constants.

Q: How do I solve a cubic equation?

A: There are several methods to solve a cubic equation, including factoring, grouping, and using the quadratic formula. In our previous article, we used the method of factoring to solve the equation ${x^3 + 27x^2 + 206x + 480 = 0}$.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations of the form ${ax^2 + bx + c = 0}$. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: How do I use the quadratic formula to solve a cubic equation?

A: While the quadratic formula is typically used to solve quadratic equations, it can also be used to solve cubic equations by first factoring the equation and then using the quadratic formula to solve the resulting quadratic equation.

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

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

• Not factoring the equation correctly
• Not using the correct method to solve the equation
• Not checking for extraneous solutions
• Not simplifying the equation correctly

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you can plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

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

A: Solving cubic equations has 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 computer graphics and game development to create realistic models of objects and environments.

Q: Can you provide some examples of cubic equations?

A: Yes, here are some examples of cubic equations:

• ${x^3 + 2x^2 - 7x - 12 = 0}$
• ${x^3 - 4x^2 + 5x + 6 = 0}$
• ${x^3 + 3x^2 - 2x - 1 = 0}$

Conclusion

In conclusion, solving cubic equations is an important topic in mathematics and has many real-world applications. By understanding the methods and techniques used to solve cubic equations, you can apply them to a wide range of problems and challenges.

Final Answer

The final answer is: -36, -170