Solving 3y³ + 9y² - 162y = 0 A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of algebra to tackle a fascinating problem: solving the cubic equation 3y³ + 9y² - 162y = 0. Cubic equations, with their highest power of 3, might seem intimidating at first, but don't worry, we'll break it down step by step. Our goal is not just to find the solutions for y but also to understand the underlying principles and techniques involved. This will not only help you solve this particular equation but also equip you with the skills to handle similar problems in the future. We'll use a combination of factoring, the quadratic formula, and a bit of algebraic manipulation to unravel this equation and discover its roots. So, grab your pencils and let's get started on this exciting mathematical journey!

Step 1: Simplifying the Equation by Factoring Out the Greatest Common Factor

The first step in solving any equation, especially one like 3y³ + 9y² - 162y = 0, is to look for common factors. Factoring is a powerful technique that simplifies the equation, making it easier to handle. In this case, we can see that each term has a common factor of 3y. Factoring out 3y from the equation, we get:

3y(y² + 3y - 54) = 0

This step significantly simplifies the equation. Instead of dealing with a cubic polynomial directly, we now have a product of 3y and a quadratic expression (y² + 3y - 54) that equals zero. This is a crucial point because if the product of two factors is zero, then at least one of the factors must be zero. This gives us our first potential solution: 3y = 0, which implies y = 0. So, we've already found one solution! Now, we need to focus on the quadratic expression to find the remaining solutions. The beauty of factoring is that it transforms a complex problem into simpler, manageable parts. By identifying and extracting the greatest common factor, we've taken a significant step towards solving the original cubic equation. This approach not only makes the equation easier to solve but also highlights the importance of observation and simplification in mathematics.

Step 2: Solving the Quadratic Equation y² + 3y - 54 = 0 by Factoring

Now that we've factored out the greatest common factor, we're left with the quadratic equation y² + 3y - 54 = 0. To solve this, we'll attempt to factor the quadratic expression itself. Factoring a quadratic involves finding two binomials that, when multiplied together, give us the original quadratic expression. We're looking for two numbers that multiply to -54 (the constant term) and add up to 3 (the coefficient of the y term). Think of it as a puzzle: we need to find the right pieces that fit together perfectly.

After some thought, we can identify the numbers 9 and -6. Notice that 9 * (-6) = -54 and 9 + (-6) = 3. These are exactly the numbers we need! So, we can rewrite the quadratic equation as:

(y + 9)(y - 6) = 0

This factorization is the key to finding the remaining solutions. Just like before, if the product of two factors is zero, at least one of them must be zero. This gives us two more equations to solve:

• y + 9 = 0
• y - 6 = 0

Solving these simple linear equations gives us y = -9 and y = 6. So, we've successfully factored the quadratic and found two more solutions for our original cubic equation. Factoring quadratics is a fundamental skill in algebra, and mastering it opens doors to solving a wide range of problems. By breaking down the quadratic expression into its binomial factors, we've made it easy to identify the values of y that make the equation true. This method showcases the power of factoring in transforming complex expressions into simpler, solvable forms.

Step 3: Summarizing the Solutions for y

We've reached the final step in our journey to solve the cubic equation 3y³ + 9y² - 162y = 0! Let's take a moment to recap what we've done. We started by factoring out the greatest common factor, 3y, which simplified the equation to 3y(y² + 3y - 54) = 0. This immediately gave us one solution: y = 0. Then, we tackled the quadratic expression y² + 3y - 54 = 0 by factoring it into (y + 9)(y - 6) = 0. This led us to the other two solutions: y = -9 and y = 6.

Therefore, the solutions to the original cubic equation are:

• y = 0
• y = -9
• y = 6

We have successfully found all three roots of the equation! Cubic equations, being polynomials of degree three, can have up to three solutions. In this case, we found three distinct real solutions. It's important to note that not all cubic equations have three real solutions; some may have one real solution and two complex solutions. However, in this particular problem, the solutions were all real and relatively straightforward to find using factoring techniques.

This process demonstrates the power of algebraic manipulation and factoring in solving polynomial equations. By breaking down the problem into smaller, manageable parts, we were able to systematically find all the solutions. This approach not only solves the equation but also provides a deeper understanding of the underlying mathematical principles. Remember, practice is key to mastering these techniques, so keep exploring and solving different types of equations!

Alternative Method: Using the Quadratic Formula

While we successfully solved the quadratic equation by factoring, it's worth mentioning an alternative method: the quadratic formula. The quadratic formula is a powerful tool that can be used to solve any quadratic equation of the form ax² + bx + c = 0. The formula is:

y = (-b ± √(b² - 4ac)) / (2a)

In our case, the quadratic equation is y² + 3y - 54 = 0, so a = 1, b = 3, and c = -54. Plugging these values into the quadratic formula, we get:

y = (-3 ± √(3² - 4 * 1 * -54)) / (2 * 1)

Let's simplify this step by step:

y = (-3 ± √(9 + 216)) / 2

y = (-3 ± √225) / 2

y = (-3 ± 15) / 2

This gives us two possible solutions:

• y = (-3 + 15) / 2 = 12 / 2 = 6
• y = (-3 - 15) / 2 = -18 / 2 = -9

As you can see, the quadratic formula gives us the same solutions we found by factoring: y = 6 and y = -9. The quadratic formula is particularly useful when the quadratic expression is difficult or impossible to factor. It provides a reliable method for finding the solutions, regardless of the complexity of the coefficients. While factoring is often quicker and more intuitive when it's possible, the quadratic formula is a valuable tool to have in your mathematical arsenal. It's a testament to the beauty of mathematics that there are often multiple paths to the same solution, each with its own strengths and applications.

Conclusion: Mastering Cubic Equations and Beyond

So, there you have it! We've successfully solved the cubic equation 3y³ + 9y² - 162y = 0 using a combination of factoring and the quadratic formula. We found the three solutions to be y = 0, y = -9, and y = 6. This journey through the world of cubic equations has highlighted several important mathematical concepts and techniques:

• Factoring: Recognizing and extracting common factors, as well as factoring quadratic expressions, is a fundamental skill in algebra. It simplifies equations and makes them easier to solve.
• The Zero Product Property: The principle that if the product of two or more factors is zero, then at least one of the factors must be zero is crucial for solving factored equations.
• The Quadratic Formula: This powerful tool provides a reliable method for solving any quadratic equation, even those that are difficult to factor.
• Systematic Problem-Solving: Breaking down a complex problem into smaller, manageable steps is a key strategy in mathematics and in life.

By mastering these concepts, you'll be well-equipped to tackle not only cubic equations but also a wide range of other mathematical challenges. Remember, the key to success in mathematics is practice, perseverance, and a willingness to explore different approaches. Don't be afraid to make mistakes; they're valuable learning opportunities. Keep practicing, keep exploring, and most importantly, keep enjoying the beauty and power of mathematics! This problem serves as a great example of how different algebraic techniques can be combined to solve a complex equation. It also underscores the importance of having multiple tools in your mathematical toolkit, as different methods may be more suitable for different problems. So, keep honing your skills, and you'll be able to conquer any equation that comes your way!