Solving Cubic Equations Find Real Solutions To -7s² - 27s = -3s³ - 63
Introduction to Solving Cubic Equations
In the realm of algebra, solving equations is a fundamental skill, and among these, cubic equations can present a unique challenge. Cubic equations, characterized by the highest power of the variable being three, often require a blend of algebraic manipulation, factoring techniques, and a keen eye for patterns. This article delves into the process of finding all real solutions to the equation -7s² - 27s = -3s³ - 63. We will explore the necessary steps, from rearranging the equation into a standard form to employing various methods to pinpoint the real roots. Understanding these techniques is crucial not only for academic success but also for applications in various fields such as physics, engineering, and computer science. Whether you're a student grappling with algebra or simply someone with a penchant for mathematical puzzles, this exploration will offer a comprehensive understanding of how to tackle cubic equations.
Transforming the Equation into Standard Form
To effectively solve the equation -7s² - 27s = -3s³ - 63, the initial step is to rearrange it into a standard form. This involves moving all terms to one side of the equation, setting it equal to zero. This standard form, known as the general form of a cubic equation, is expressed as ax³ + bx² + cx + d = 0. By bringing all the terms to one side, we create a structure that is conducive to various solving techniques, such as factoring or using the rational root theorem. This transformation not only simplifies the equation visually but also aligns it with established methods for finding roots.
Specifically, for our equation, we begin by adding 3s³ to both sides and adding 63 to both sides. This gives us 3s³ - 7s² - 27s + 63 = 0. This form immediately reveals the coefficients of the cubic, quadratic, linear, and constant terms, making it easier to apply factoring strategies or other algebraic methods. The importance of this step cannot be overstated, as it lays the groundwork for a systematic approach to finding the solutions. Furthermore, the standard form helps in identifying potential rational roots using the Rational Root Theorem, which we may explore later in the solution process. Thus, transforming the equation into standard form is not just a preliminary step; it is a crucial maneuver that paves the way for a successful resolution.
Factoring by Grouping: A Strategic Approach
One of the most powerful techniques for solving polynomial equations, including cubics, is factoring. Factoring involves expressing the polynomial as a product of simpler polynomials. When dealing with a four-term polynomial like 3s³ - 7s² - 27s + 63 = 0, factoring by grouping can be a particularly effective strategy. This method hinges on identifying common factors within pairs of terms and then extracting those factors to simplify the expression.
To apply factoring by grouping to our equation, we first group the terms: (3s³ - 7s²) + (-27s + 63). From the first group, we can factor out s², leaving us with s²(3s - 7). From the second group, we can factor out -9, yielding -9(3s - 7). Notice that both groups now share a common factor of (3s - 7). This is a crucial observation, as it allows us to further factor the entire expression. By factoring out (3s - 7), we obtain (3s - 7)(s² - 9) = 0. The equation is now in a partially factored form, significantly closer to its solutions. The next step involves recognizing that (s² - 9) is a difference of squares, a pattern that can be factored further, simplifying the equation even more. Factoring by grouping, therefore, is not just a mechanical process; it requires keen pattern recognition and strategic application of algebraic principles.
Recognizing and Applying the Difference of Squares Pattern
After employing factoring by grouping, our equation has been simplified to (3s - 7)(s² - 9) = 0. The second factor, (s² - 9), exhibits a well-known algebraic pattern: the difference of squares. The difference of squares pattern states that a² - b² can be factored into (a + b)(a - b). Recognizing and applying this pattern is a crucial step in further simplifying the equation and revealing its roots.
In our case, s² - 9 fits the difference of squares pattern perfectly, where s² corresponds to a² and 9 corresponds to b². Since 9 is 3², we can rewrite (s² - 9) as (s² - 3²). Applying the difference of squares factorization, we get (s + 3)(s - 3). Substituting this back into our equation, we now have (3s - 7)(s + 3)(s - 3) = 0. This fully factored form is incredibly powerful because it directly leads us to the solutions of the equation. Each factor represents a potential root, and by setting each factor equal to zero, we can easily isolate the values of s that satisfy the equation. The recognition and application of the difference of squares pattern not only simplifies the algebraic manipulation but also showcases the interconnectedness of various factoring techniques in solving polynomial equations. This skill is invaluable in a wide range of mathematical contexts, from solving simple quadratics to more complex polynomial problems.
Identifying Real Solutions from Factored Form
With the equation fully factored as (3s - 7)(s + 3)(s - 3) = 0, the final step in finding the real solutions is to apply the Zero Product Property. This fundamental property states that if the product of several factors is zero, then at least one of the factors must be zero. By setting each factor in our equation equal to zero, we create a set of simpler equations that can be solved individually to find the roots of the original cubic equation.
Starting with the first factor, we set 3s - 7 = 0. Solving for s, we add 7 to both sides, yielding 3s = 7. Then, dividing both sides by 3, we find the first solution: s = 7/3. Next, we consider the second factor, s + 3 = 0. Subtracting 3 from both sides gives us the second solution: s = -3. Finally, we set the third factor, s - 3 = 0. Adding 3 to both sides provides the third solution: s = 3. Therefore, the real solutions to the equation -7s² - 27s = -3s³ - 63 are s = 7/3, s = -3, and s = 3. These values represent the points where the cubic function intersects the s-axis on a graph, and they fully satisfy the original equation. Identifying the real solutions from the factored form is a direct application of the Zero Product Property and underscores the importance of factoring in solving polynomial equations.
Verification of Solutions
To ensure the accuracy of our solutions, it is always a prudent step to verify them by substituting each value back into the original equation. This process helps catch any potential errors made during the algebraic manipulation and confirms that the solutions indeed satisfy the given equation. In the context of problem-solving, verification is not merely a formality; it is an integral part of the solution process that adds a layer of confidence to the final answer.
Let's start by verifying s = 7/3. Substituting this value into the original equation, -7s² - 27s = -3s³ - 63, we have -7(7/3)² - 27(7/3) = -3(7/3)³ - 63. Evaluating both sides, we get -7(49/9) - 63 = -3(343/27) - 63, which simplifies to -343/9 - 63 = -343/9 - 63. Since both sides are equal, s = 7/3 is indeed a valid solution. Next, we verify s = -3. Substituting this into the original equation, we have -7(-3)² - 27(-3) = -3(-3)³ - 63, which simplifies to -7(9) + 81 = -3(-27) - 63, further simplifying to -63 + 81 = 81 - 63, which gives us 18 = 18. This confirms that s = -3 is also a valid solution. Lastly, we verify s = 3. Substituting this into the original equation, we get -7(3)² - 27(3) = -3(3)³ - 63, which simplifies to -7(9) - 81 = -3(27) - 63, further simplifying to -63 - 81 = -81 - 63, which gives us -144 = -144. This confirms that s = 3 is a valid solution. Since all three solutions, s = 7/3, s = -3, and s = 3, satisfy the original equation, we can confidently conclude that these are indeed the real solutions to the equation. This verification process underscores the importance of meticulousness and thoroughness in solving mathematical problems.
Conclusion
In conclusion, we have successfully navigated the process of finding all real solutions to the cubic equation -7s² - 27s = -3s³ - 63. The journey began with transforming the equation into its standard form, 3s³ - 7s² - 27s + 63 = 0, which laid the foundation for subsequent steps. We then employed the factoring by grouping technique, which allowed us to express the equation as (3s - 7)(s² - 9) = 0. Recognizing the difference of squares pattern in the factor (s² - 9) further simplified the equation to (3s - 7)(s + 3)(s - 3) = 0. Applying the Zero Product Property, we identified the potential real solutions: s = 7/3, s = -3, and s = 3. Finally, we rigorously verified each solution by substituting it back into the original equation, confirming their validity.
This step-by-step approach highlights the significance of algebraic manipulation, pattern recognition, and the application of fundamental properties in solving cubic equations. The process not only yields the solutions but also reinforces key mathematical concepts and problem-solving strategies. The techniques discussed here, such as factoring by grouping and recognizing the difference of squares, are applicable to a wide range of algebraic problems, making this exploration a valuable exercise in mathematical thinking. Ultimately, solving cubic equations involves a blend of algebraic skill, strategic thinking, and careful execution, culminating in a satisfying resolution.
