Solving Cubic Equations: Factoring And Quadratic Formula
Hey there, math enthusiasts! Today, we're diving into the world of polynomial equations, specifically tackling a cubic equation. Our goal is to solve the equation x³ - 29x² + 54x = 0 and pinpoint all the solutions. We'll be using two powerful tools: factoring and, if needed, the quadratic formula. Buckle up, because we're about to unravel this mathematical mystery together! This exploration is designed to be clear, concise, and easy to follow, even if you're just brushing up on your algebra skills or tackling this concept for the first time. The beauty of mathematics lies in its logical progression, and solving equations like these is a testament to that. Let's get started!
Step-by-Step Solution: Unveiling the Roots
Let's meticulously solve the equation x³ - 29x² + 54x = 0. Our approach will be a combination of careful observation, strategic factoring, and potentially, a little help from the quadratic formula. Remember, the ultimate goal is to find the values of x that make the equation true. These values are often referred to as the roots or solutions of the equation. Finding these roots allows us to fully understand the behavior of the polynomial and its graphical representation. Polynomial equations are fundamental in many areas of mathematics and science, and mastering their solution is a valuable skill.
Firstly, we should always look for the simplest method of solving the equation. Notice that each term in the equation has an x. This immediately hints at the possibility of factoring out an x. By doing this, we simplify the equation significantly and make it easier to work with. The process of factoring helps break down the equation into smaller, more manageable parts, allowing us to find the roots more efficiently. Factoring is like dissecting a puzzle, where each piece contributes to the final solution. The initial factorization is a crucial step because it reduces the degree of the polynomial, potentially leading to a simpler equation. Let's do this first step. Factor out x: x(x² - 29x + 54) = 0. Now, we have successfully factored out an x from the original equation. This tells us immediately that x = 0 is one of the solutions. This comes directly from the factored form of the equation: if x = 0, the entire expression equals zero. This is our first root! Identifying the first root is like finding the first clue to solving a more intricate puzzle. This step lays the groundwork for further simplification and helps us proceed methodically towards our goal of finding all the solutions. Next, we turn our attention to the quadratic expression inside the parentheses: x² - 29x + 54. Now that we have one root in our pocket, we can try to factor this expression. If the quadratic expression is factorable, this will lead us to the remaining solutions easily. The factored quadratic will then be multiplied by the x we factored in the previous step, which leads to a product of factors, each representing a potential solution.
Now, we're looking for two numbers that multiply to 54 and add up to -29. This is where a bit of trial and error, or simply recalling your multiplication tables, comes in handy. It's often helpful to list the factors of 54 to narrow down the possibilities. We need to consider both positive and negative factors, given that the sum we're looking for is negative. Factoring quadratic expressions is a fundamental skill in algebra, enabling you to rewrite the expressions in a product of simpler factors. It's essentially the reverse process of multiplying binomials. This helps simplify the expression, allowing you to quickly spot the roots of the equation. After considering different factor pairs, we find that -2 and -27 are the two numbers that multiply to 54 and sum to -29. Thus, we can factor the quadratic expression to (x - 2)(x - 27). This step is about splitting the middle term of the quadratic expression. It is like taking apart a structure to see how it's built or finding components.
Putting it all together, our fully factored equation is x(x - 2)(x - 27) = 0. For this equation to be true, any of the factors can be equal to zero. This principle is fundamental in solving factored polynomial equations. Because the product of the factors equals zero, the values that make each factor zero are the solutions of the equation. This is the essence of solving factored equations. Therefore, we can set each factor to zero to find the solutions. The values that make each factor zero are the roots of our equation. It is also an important step to ensure we do not miss any possible solutions. Setting x = 0 gives us our first solution. Setting (x-2) = 0 implies x = 2, which gives us our second solution. Setting (x-27) = 0 implies x = 27, which gives us our third solution. So, the solutions to the equation are x = 0, x = 2, and x = 27. We did it!
Verification of the Solutions: Checking Our Work
To ensure we've found all the correct solutions, let's substitute each solution back into the original equation and verify that it satisfies the equation. Verification is an important step. It is like double-checking our calculations in order to avoid mistakes. It is an important quality of a responsible scientist or mathematician to verify that the answer is accurate. We will start by substituting x = 0 into the original equation: (0)³ - 29(0)² + 54(0) = 0. This simplifies to 0 = 0, confirming that x = 0 is indeed a solution. Our first solution checks out! We did it! Next, we will substitute x = 2 into the original equation: (2)³ - 29(2)² + 54(2) = 0. Calculating this expression, we have 8 - 116 + 108 = 0. This simplifies to 0 = 0, confirming that x = 2 is also a solution. This is our second success! Finally, we substitute x = 27 into the original equation: (27)³ - 29(27)² + 54(27) = 0. This calculation leads to 19683 - 21069 + 1458 = 0, which again simplifies to 0 = 0. This final verification confirms that x = 27 is indeed a solution. Our entire solution is correct!
Conclusion: Solutions Unveiled
We successfully solved the cubic equation x³ - 29x² + 54x = 0 using factoring techniques. The solutions to the equation are x = 0, x = 2, and x = 27. The process involved factoring out a common term and then factoring the resulting quadratic expression. We then verified our solutions by substituting them back into the original equation. This approach highlights the power of factoring in solving polynomial equations. The ability to factor is a fundamental skill in algebra and is essential for solving many types of equations. Knowing how to factor makes solving equations much easier. We broke down the problem into smaller, simpler pieces, which made the complex problem more manageable. Mastery of factoring and applying the quadratic formula empowers you to tackle a wide variety of polynomial equations. Keep practicing, and you'll find that solving polynomial equations becomes increasingly intuitive! Keep going with the exercises, and you will become a master of these equations.
Now you're equipped to solve similar cubic equations! Keep practicing, and you'll become more confident in your ability to solve them. Remember, the more you practice, the easier it becomes! The combination of factoring and using the quadratic formula is a great method to solve these problems. Congratulations on your understanding of solving cubic equations!
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