Solve For X: Unveiling The Equation $1,331x^3 - 216 = 0$

Nov 14, 2025 by Alex Johnson 57 views

Are you ready to dive into the world of algebra and uncover the secrets hidden within the equation $1,331x^3 - 216 = 0$ ? This seemingly complex equation is actually a fantastic opportunity to sharpen your problem-solving skills and gain a deeper understanding of algebraic principles. In this comprehensive guide, we'll break down the equation step by step, making sure you grasp every concept along the way. Whether you're a seasoned math enthusiast or just starting your algebraic journey, this exploration promises to be both informative and engaging. Let's embark on this mathematical adventure together!

Understanding the Basics of Cubic Equations

Before we jump into the equation, it's helpful to understand what we're dealing with. The equation $1,331x^3 - 216 = 0$ is a cubic equation, which means it involves a variable raised to the power of 3. Cubic equations can have up to three solutions, also known as roots. These roots can be real numbers or complex numbers. In our case, we're looking for the real solutions, the values of x that satisfy the equation. This is where we will use some formula and simple mathematic rules to find the values of x. The equation will be simplified so that all of us can understand it easily. Understanding the basics will make solving the equation much easier.

Cubic equations might seem intimidating at first, but with a solid grasp of the fundamentals, they become much more approachable. The key is to recognize the patterns and apply the appropriate techniques. One of the most common methods for solving cubic equations involves factoring. Factoring is the process of breaking down an expression into its components. For our equation, we'll try to factor it using the difference of cubes formula, which states that $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ .

Knowing how to factor is essential, as it allows us to simplify the equation and isolate the variable. Another helpful concept is the zero-product property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property is crucial for finding the solutions to our factored equation. Furthermore, it's always beneficial to review the basics of exponents and roots. Understanding the relationship between cubes and cube roots will be critical in solving for x. Remembering that the cube root of a number is the value that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2, because 2 * 2 * 2 = 8. Finally, don't forget the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that we perform the calculations in the correct sequence, leading to the correct solution. By having these concepts in mind, we can effectively navigate the cubic equation and arrive at the correct answer.

Step-by-Step Solution to the Equation

Now, let's roll up our sleeves and solve the equation $1,331x^3 - 216 = 0$ . The goal is to isolate x and find its value. Here's a detailed walkthrough:

Rewrite the Equation: The first step is to rewrite the equation in a form that makes it easier to factor. Notice that both 1,331 and 216 are perfect cubes. Specifically, $1,331 = 11^3$ and $216 = 6^3$ . So, we can rewrite the equation as $(11x)^3 - 6^3 = 0$ .
Apply the Difference of Cubes Formula: Now, we can use the difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ . In our case, $a = 11x$ and $b = 6$ . Applying the formula, we get: $(11x - 6)((11x)^2 + (11x)(6) + 6^2) = 0$ .
Simplify the Equation: Let's simplify the equation further: $(11x - 6)(121x^2 + 66x + 36) = 0$ .
Apply the Zero-Product Property: According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This means either $11x - 6 = 0$ or $121x^2 + 66x + 36 = 0$ .
Solve the Linear Equation: Let's solve the first equation: $11x - 6 = 0$ . Add 6 to both sides: $11x = 6$ . Divide both sides by 11: $x = rac{6}{11}$ .
Solve the Quadratic Equation: Now, let's solve the second equation: $121x^2 + 66x + 36 = 0$ . This is a quadratic equation. We can try to factor it or use the quadratic formula. In this case, we can observe that the quadratic expression is a perfect square trinomial, specifically $(11x + 6)^2 = 0$ . Taking the square root of both sides, we get $11x + 6 = 0$ . Subtract 6 from both sides: $11x = -6$ . Divide both sides by 11: $x = - rac{6}{11}$ .
Identify the Solution: Therefore, the solutions to the equation are $x = rac{6}{11}$ and $x = - rac{6}{11}$ .

Choosing the Correct Answer

Now that we've solved the equation and found the solutions for x, let's revisit the answer choices provided. Remember, the original question presented four possible values for x. The answer choices were:

A. $- rac{6}{11}$
B. $rac{11}{6}$
C. $rac{1}{6}$
D. $rac{6}{11}$

From our calculations, we have determined that the values of x that satisfy the equation $1,331x^3 - 216 = 0$ are $rac{6}{11}$ and $- rac{6}{11}$ . Considering the answer choices, we see that options A and D match our findings. Thus, both A and D are correct, however in the question, we are asked to pick the correct value of x. Therefore, there can only be one correct answer, and that is either $x = rac{6}{11}$ or $x = - rac{6}{11}$ . In a multiple-choice question, we would ideally expect only one correct answer, to avoid confusion.

In multiple-choice questions, it is crucial to carefully examine all the options and select the one that aligns with the solution. In this case, since the question requires us to pick one correct value of x, both the value are valid answers, so we can pick either value and it will be correct.

Conclusion: Mastering the Cubic Equation

Congratulations! You've successfully navigated the equation $1,331x^3 - 216 = 0$ and found the values of x. By breaking down the equation step by step, we've demonstrated how to apply key algebraic principles like factoring, the difference of cubes, and the zero-product property. Remember, practice is key to mastering algebraic equations. The more you work through problems, the more confident you'll become in your problem-solving abilities. Don't hesitate to revisit the steps, practice with similar equations, and explore additional resources. With each equation you solve, you'll be building a stronger foundation in algebra and expanding your mathematical horizons. Keep up the great work, and enjoy the journey of learning and discovery! You've now equipped yourself with the knowledge to tackle similar cubic equations and strengthen your problem-solving skills. So keep practicing, stay curious, and continue exploring the fascinating world of mathematics!

For more practice and in-depth explanations, check out these resources:

Khan Academy: https://www.khanacademy.org/