When The Product Of 6 And The Square Of A Number Is Increased By 5 Times The Number, The Result Is 4

Let's delve into the problem-solving process for this intriguing mathematical puzzle. Our primary objective is to identify the values that satisfy the given condition: when the product of 6 and the square of a number is increased by 5 times the number, the result is 4. This problem necessitates a methodical approach, incorporating algebraic techniques to decipher the unknown values. The solution involves formulating a quadratic equation and then solving it using appropriate methods, such as factoring, completing the square, or employing the quadratic formula. By carefully analyzing the equation and its solutions, we can pinpoint the specific values that align with the initial problem statement. Our journey begins with translating the word problem into a mathematical expression, laying the groundwork for a clear and concise solution. Understanding the fundamental principles of algebra is paramount in tackling such challenges, enabling us to transform abstract concepts into concrete results. This process not only enhances our problem-solving skills but also deepens our comprehension of mathematical relationships. The beauty of mathematics lies in its ability to provide definitive answers through logical reasoning and precise calculations. As we navigate this problem, we'll uncover the elegance and power of algebraic methods in resolving real-world scenarios. The quest for the solution is not merely about finding the numbers; it's about embracing the analytical journey and appreciating the interconnectedness of mathematical ideas. By dissecting the problem step by step, we gain a profound understanding of how mathematical principles operate, paving the way for future explorations and discoveries.

Transforming the Word Problem into an Equation

To effectively tackle this problem, we must first translate the word problem into a mathematical equation. This crucial step allows us to manipulate the given information using algebraic techniques and ultimately solve for the unknown values. The problem states: "When the product of 6 and the square of a number is increased by 5 times the number, the result is 4." Let's break this down piece by piece. We'll represent the unknown number with the variable x. The square of the number is then x², and the product of 6 and the square of the number is 6x². Next, 5 times the number is simply 5x. The problem tells us that when 6x² is increased by 5x, the result is 4. This translates to addition, so we have 6x² + 5x. Finally, the problem states that this entire expression equals 4. Therefore, our equation is: 6x² + 5x = 4. This equation is a quadratic equation, a type of polynomial equation that has the general form ax² + bx + c = 0. To solve it, we need to rearrange it into this standard form. We can do this by subtracting 4 from both sides of the equation: 6x² + 5x - 4 = 0. Now we have a quadratic equation in standard form, ready to be solved. The coefficients are a = 6, b = 5, and c = -4. With this equation, we can now employ various methods to find the values of x that satisfy the given condition. Solving this quadratic equation is the key to unlocking the solution to our original word problem. By understanding how to translate verbal descriptions into mathematical expressions, we empower ourselves to tackle a wide array of problems. The journey from words to equations is a fundamental skill in mathematics and opens the door to deeper analytical insights. Each step in this process, from identifying the unknowns to constructing the equation, is crucial for arriving at the correct solution. The equation we've formed is a powerful tool, representing the problem's core relationship in a concise and manageable form. By mastering the art of translation, we not only solve individual problems but also develop a stronger foundation for mathematical reasoning.

Solving the Quadratic Equation

Now that we've successfully translated the word problem into a quadratic equation, 6x² + 5x - 4 = 0, the next step is to solve for the unknown variable, x. There are several methods we can employ to solve quadratic equations, each with its own strengths and applications. The most common methods include factoring, completing the square, and using the quadratic formula. For this particular equation, factoring might seem challenging at first glance, but it's a viable option if we can find two binomials that multiply to give us the quadratic expression. The quadratic formula, on the other hand, is a universal method that works for any quadratic equation, regardless of its factorability. Let's explore the quadratic formula in detail. The quadratic formula states that for an equation of the form ax² + bx + c = 0, the solutions for x are given by: x = (-b ± √(b² - 4ac)) / (2a). In our case, a = 6, b = 5, and c = -4. Plugging these values into the formula, we get: x = (-5 ± √(5² - 4 * 6 * -4)) / (2 * 6). Simplifying the expression under the square root: x = (-5 ± √(25 + 96)) / 12. Further simplification gives: x = (-5 ± √121) / 12. Since √121 = 11, we have: x = (-5 ± 11) / 12. This yields two possible solutions for x: x₁ = (-5 + 11) / 12 = 6 / 12 = 1/2 and x₂ = (-5 - 11) / 12 = -16 / 12 = -4/3. Therefore, the two solutions to the quadratic equation are x = 1/2 and x = -4/3. These values represent the numbers that satisfy the original condition stated in the word problem. The quadratic formula provides a systematic approach to solving quadratic equations, ensuring that we find all possible solutions. By mastering this method, we equip ourselves with a powerful tool for tackling a wide range of mathematical problems. The solutions we've obtained are not just abstract numbers; they are the specific values that make the original equation true. Each step in the process, from applying the formula to simplifying the results, is crucial for arriving at the correct answers.

Verifying the Solutions

After obtaining the solutions x = 1/2 and x = -4/3 for the quadratic equation 6x² + 5x - 4 = 0, it's essential to verify these solutions. Verification ensures that the values we've calculated indeed satisfy the original equation and, by extension, the initial word problem. This step is crucial for confirming the accuracy of our work and identifying any potential errors. To verify the solutions, we substitute each value back into the original equation and check if the equation holds true. Let's start with x = 1/2. Substituting this value into the equation, we get: 6*(1/2)² + 5*(1/2) - 4 = 6*(1/4) + 5/2 - 4 = 3/2 + 5/2 - 4 = 8/2 - 4 = 4 - 4 = 0. Since the result is 0, the equation holds true for x = 1/2. Now, let's verify x = -4/3. Substituting this value into the equation, we get: 6*(-4/3)² + 5*(-4/3) - 4 = 6*(16/9) - 20/3 - 4 = 32/3 - 20/3 - 4 = 12/3 - 4 = 4 - 4 = 0. Again, the result is 0, confirming that the equation also holds true for x = -4/3. Both solutions, x = 1/2 and x = -4/3, satisfy the quadratic equation 6x² + 5x - 4 = 0. This verification process not only validates our calculations but also reinforces our understanding of the relationship between the solutions and the equation. By substituting the solutions back into the original problem context, we can ensure that they make sense in the real-world scenario. Verification is a cornerstone of mathematical problem-solving, providing a final check on our work and building confidence in our results. Each step in this process, from substituting the values to simplifying the expressions, is crucial for arriving at a reliable conclusion. The solutions we've verified are not just abstract numbers; they are the specific values that fulfill the conditions of the original problem. By mastering the art of verification, we enhance our problem-solving skills and cultivate a meticulous approach to mathematical reasoning.

Conclusion: Identifying the Correct Values

In conclusion, after translating the word problem into a quadratic equation, solving for the unknown variable, and verifying the solutions, we have successfully identified the values that satisfy the given condition. The original problem stated: "When the product of 6 and the square of a number is increased by 5 times the number, the result is 4." We transformed this statement into the equation 6x² + 5x = 4, which we then rearranged into the standard quadratic form: 6x² + 5x - 4 = 0. By applying the quadratic formula, we found two potential solutions: x = 1/2 and x = -4/3. To ensure the accuracy of our results, we meticulously verified each solution by substituting it back into the original equation. Our verification process confirmed that both x = 1/2 and x = -4/3 satisfy the equation, thus validating them as the correct values. Therefore, the numbers that meet the problem's criteria are 1/2 and -4/3. This problem-solving journey highlights the power of algebraic techniques in deciphering mathematical puzzles. From translating word problems into equations to employing the quadratic formula and verifying solutions, each step plays a crucial role in arriving at the correct answer. The process not only enhances our mathematical skills but also deepens our understanding of how mathematical principles operate. The solutions we've obtained are not just abstract numbers; they are the specific values that make the original statement true. By mastering these techniques, we equip ourselves with valuable tools for tackling a wide range of mathematical challenges. The beauty of mathematics lies in its ability to provide definitive answers through logical reasoning and precise calculations. The problem-solving journey, from initial formulation to final verification, is a testament to the elegance and effectiveness of mathematical methods. By dissecting the problem step by step, we gain a profound appreciation for the interconnectedness of mathematical ideas and the power of algebraic tools in resolving real-world scenarios.