Is (15x² - 82x + 35) Equal To (5x - 4)(3x - 14) - 21? A Detailed Analysis

Is the quadratic expression 15x² - 82x + 35 truly equivalent to the expression (5x - 4)(3x - 14) - 21? This seemingly straightforward question delves into the core principles of algebraic manipulation and simplification. To definitively answer this, we must embark on a step-by-step journey, meticulously expanding, simplifying, and comparing the two expressions. Understanding the equality of algebraic expressions is not merely an academic exercise; it forms the bedrock of countless mathematical applications, from solving equations to modeling real-world phenomena. In this comprehensive exploration, we will dissect each expression, revealing the underlying structure and determining whether they indeed represent the same mathematical entity. Our analysis will not only provide a conclusive answer but also reinforce fundamental algebraic techniques, enhancing your understanding of mathematical equivalency. So, let's dive into the realm of quadratic expressions and unravel this intriguing mathematical puzzle.

Expanding the Expression (5x - 4)(3x - 14) - 21

The cornerstone of our investigation lies in the proper expansion and simplification of the expression (5x - 4)(3x - 14) - 21. This involves a meticulous application of the distributive property, often remembered by the mnemonic FOIL (First, Outer, Inner, Last), followed by combining like terms. Let's embark on this algebraic journey step by step. Initially, we focus on expanding the product of the two binomials, (5x - 4) and (3x - 14). This requires us to multiply each term of the first binomial with each term of the second binomial, ensuring that every combination is accounted for. The first terms, 5x and 3x, are multiplied to yield 15x². Then, the outer terms, 5x and -14, are multiplied to give -70x. Moving to the inner terms, -4 and 3x, their product is -12x. Finally, the last terms, -4 and -14, are multiplied to produce +56. Now, we meticulously gather these terms: 15x², -70x, -12x, and +56. The next step is crucial: combining the like terms. In this case, the terms -70x and -12x are like terms because they both contain the variable x raised to the power of 1. Combining them, we obtain -82x. Therefore, the expanded form of the product (5x - 4)(3x - 14) is 15x² - 82x + 56. However, our task is not yet complete. We must not forget the crucial subtraction of 21 from the entire expression. Subtracting 21 from 15x² - 82x + 56 gives us 15x² - 82x + 35. This final form is the simplified representation of the expression (5x - 4)(3x - 14) - 21, ready to be compared with the other expression in question. This careful process of expansion and simplification highlights the importance of precision and attention to detail in algebraic manipulations. Any error in this process could lead to an incorrect conclusion about the equivalence of the expressions.

Analyzing the Quadratic Expression 15x² - 82x + 35

Now, let's turn our attention to the quadratic expression 15x² - 82x + 35. This expression, in its standard form, presents a clear structure that allows for direct comparison with the simplified form we obtained in the previous section. The expression consists of three terms: a quadratic term (15x²), a linear term (-82x), and a constant term (35). Each term plays a crucial role in defining the behavior and properties of the quadratic expression. The coefficient of the quadratic term, 15, dictates the parabola's curvature and concavity. The linear term, -82x, influences the parabola's slope and position on the coordinate plane. And finally, the constant term, 35, determines the parabola's y-intercept – the point where the parabola intersects the y-axis. Examining this expression in isolation, we can discern its basic characteristics. It is a quadratic expression, meaning it will graph as a parabola. The positive coefficient of the x² term indicates that the parabola opens upwards. The coefficients themselves are specific values, and they are crucial in determining whether this expression is equivalent to another expression. There are no further simplifications possible for this expression in its current form; it is already in its simplest polynomial form. This means we are ready to directly compare it to the simplified form of the other expression to determine their equivalence. The analysis of this expression underscores the importance of recognizing the structure and components of polynomial expressions, as this understanding is fundamental to algebraic comparisons and manipulations. Each term contributes to the overall behavior of the expression, and careful consideration of these terms is essential for accurate mathematical analysis.

Comparing the Simplified Expressions

The moment of truth has arrived – we must now compare the simplified forms of the two expressions to determine their equivalence. On one hand, we have the quadratic expression 15x² - 82x + 35. This expression, as we analyzed, is in its simplest form, with a clear quadratic, linear, and constant term. On the other hand, we have the simplified form of (5x - 4)(3x - 14) - 21, which, after meticulous expansion and simplification, resulted in 15x² - 82x + 35. The comparison is now strikingly straightforward. We have two expressions, each represented in its simplified polynomial form. By visually inspecting the two expressions, we can observe a remarkable congruence: the coefficients of the corresponding terms are identical. The quadratic term in both expressions is 15x². The linear term in both expressions is -82x. And, crucially, the constant term in both expressions is 35. This precise match across all terms leads us to a definitive conclusion: the two expressions are indeed equivalent. This equivalence is not merely a numerical coincidence; it signifies that for any value of x that we substitute into either expression, the result will be the same. This fundamental property of equivalent expressions is a cornerstone of algebra and mathematical reasoning. The significance of this finding extends beyond this specific problem. It highlights the power of algebraic manipulation and simplification as tools for revealing underlying relationships between seemingly different expressions. Recognizing and establishing equivalences is a crucial skill in mathematics, enabling us to solve equations, simplify complex problems, and gain deeper insights into mathematical structures. Our comparison underscores the importance of careful, step-by-step algebraic procedures in achieving accurate results and uncovering mathematical truths.

Conclusion: The Expressions are Equivalent

In conclusion, after a thorough and meticulous examination, we can definitively state that the expression 15x² - 82x + 35 is indeed equivalent to the expression (5x - 4)(3x - 14) - 21. This determination was achieved through a systematic process of expanding and simplifying the second expression, followed by a direct comparison with the first expression. The key to our success lay in the careful application of algebraic principles, particularly the distributive property and the combination of like terms. By expanding the product of the binomials (5x - 4)(3x - 14) and subsequently subtracting 21, we arrived at the simplified form 15x² - 82x + 35. This simplified form was then directly compared to the original expression, 15x² - 82x + 35, revealing a perfect match. The identical nature of the two expressions, term by term, leaves no room for doubt – they are mathematically equivalent. This equivalence has profound implications. It signifies that for any value assigned to the variable x, both expressions will yield the same numerical result. This property is a hallmark of equivalent algebraic expressions and is fundamental to many mathematical operations, including solving equations and simplifying complex formulas. The process we undertook to establish this equivalence underscores the importance of algebraic manipulation and simplification in mathematics. These techniques allow us to transform expressions into more manageable forms, reveal hidden relationships, and ultimately gain a deeper understanding of mathematical structures. This exploration serves as a powerful reminder of the elegance and precision of algebraic reasoning, and it reinforces the vital role that equivalence plays in the broader landscape of mathematics. Understanding and identifying equivalent expressions is not just a theoretical exercise; it is a practical skill that empowers us to solve problems, make connections, and appreciate the beauty of mathematical relationships.

Further Exploration: The Significance of Equivalent Expressions

Beyond the specific example we've explored, the concept of equivalent expressions holds immense significance in the broader realm of mathematics. Understanding and manipulating equivalent expressions is a cornerstone of algebra and crucial for tackling more advanced mathematical concepts. Equivalent expressions, at their core, represent the same mathematical relationship, even if they appear different superficially. This means that they produce the same output for any given input value. This property is incredibly valuable in various mathematical contexts. In equation solving, for instance, we often manipulate one side of an equation by substituting it with an equivalent expression. This allows us to isolate the variable and find its value without altering the equation's solution. Simplification is another crucial application. Complex expressions can often be simplified into more manageable forms by identifying and applying equivalencies. This makes the expressions easier to work with and understand. For instance, factoring a quadratic expression into the product of two binomials is a prime example of using equivalence for simplification. Equivalent expressions are also vital in calculus and other advanced mathematical fields. In calculus, simplifying expressions before differentiation or integration can significantly reduce the complexity of the calculations. Recognizing equivalencies can lead to more elegant and efficient solutions. The ability to identify and manipulate equivalent expressions also fosters a deeper understanding of mathematical relationships. It allows us to see how different expressions can represent the same underlying concept, providing a more holistic view of mathematics. Furthermore, the concept of equivalence extends beyond algebraic expressions. It applies to trigonometric expressions, logarithmic expressions, and even more abstract mathematical structures. Mastering the principles of equivalence is therefore a fundamental step in mathematical proficiency. It empowers us to approach problems with greater flexibility, efficiency, and understanding. As we continue our mathematical journey, the ability to recognize and utilize equivalent expressions will undoubtedly prove to be an invaluable asset.

Keywords: algebraic expressions, quadratic expression, equivalent expressions, simplification, expansion, distributive property, FOIL, combining like terms, mathematical equivalence, equation solving