Simplifying The Expression -3(y-5)^2-9+7y

In the realm of mathematics, expressions often present themselves as intricate puzzles, demanding careful dissection and strategic simplification. Today, we embark on a journey to unravel the expression -3(y-5)^2-9+7y, a seemingly complex arrangement of numbers, variables, and operations. Our mission is twofold: to meticulously dissect the process of simplification and to arrive at the final, simplified product. Along the way, we will explore the underlying principles that govern mathematical manipulations and uncover the elegance hidden within this algebraic tapestry.

Dissecting the Expression: A Step-by-Step Approach

Our initial task is to break down the expression into manageable components, addressing each operation in a logical sequence. The expression at hand, -3(y-5)^2-9+7y, presents us with a combination of exponents, parentheses, multiplication, and addition. To navigate this mathematical landscape effectively, we must adhere to the established order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

1. Tackling the Parentheses: Our first port of call is the expression nestled within the parentheses: (y-5). This term represents a binomial, a two-term expression, and it forms the foundation for our subsequent operations. However, before we can proceed further, we must address the exponent that looms outside the parentheses.
2. Conquering the Exponent: The exponent of 2 signifies that we must square the binomial (y-5). This involves multiplying the binomial by itself: (y-5)(y-5). To execute this multiplication, we employ the distributive property, ensuring that each term within the first binomial is multiplied by each term within the second binomial. This yields: y^2 - 5y - 5y + 25, which simplifies to y^2 - 10y + 25. This trinomial, a three-term expression, now replaces the original (y-5)^2 term in our expression.
3. Navigating the Multiplication: With the exponent resolved, we turn our attention to the multiplication operation. The term -3 is poised to multiply the trinomial we just obtained: -3(y^2 - 10y + 25). Again, we invoke the distributive property, ensuring that -3 is multiplied by each term within the trinomial. This results in: -3y^2 + 30y - 75. This new expression now takes its place in our ongoing simplification process.
4. Embracing Addition and Subtraction: The final stage of our journey involves the addition and subtraction operations. We now have the expression: -3y^2 + 30y - 75 - 9 + 7y. Our task is to combine like terms, those terms that share the same variable and exponent. In this case, we have two terms involving the variable 'y': 30y and 7y. Adding these together, we get 37y. Similarly, we have two constant terms: -75 and -9. Combining these, we arrive at -84.

Thus, after meticulously following the order of operations and combining like terms, we arrive at the simplified product: -3y^2 + 37y - 84. This elegant expression represents the culmination of our algebraic expedition, a testament to the power of methodical simplification.

Verifying the Simplified Product: Unveiling the Truth

Now that we have arrived at the simplified product, it is crucial to verify its accuracy. To achieve this, we will explore two distinct yet complementary approaches:

Method 1: Back-Substitution

Back-substitution involves substituting a numerical value for the variable 'y' in both the original expression and the simplified product. If the two expressions yield the same result, our simplification is likely correct. Let us choose a convenient value for 'y', such as y = 2. Substituting this value into the original expression, we get:

-3(2-5)^2 - 9 + 7(2) = -3(-3)^2 - 9 + 14 = -3(9) - 9 + 14 = -27 - 9 + 14 = -22

Now, let us substitute y = 2 into our simplified product:

-3(2)^2 + 37(2) - 84 = -3(4) + 74 - 84 = -12 + 74 - 84 = -22

Since both expressions yield the same result (-22), our simplified product holds strong.

Method 2: Step-by-Step Verification

This method involves meticulously reviewing each step of our simplification process, ensuring that no errors have crept in along the way. We retrace our steps, scrutinizing each operation and each combination of terms. This methodical approach allows us to identify any potential missteps and rectify them before they lead us astray.

By employing both back-substitution and step-by-step verification, we can confidently assert the accuracy of our simplified product. The expression -3y^2 + 37y - 84 stands as a testament to the power of careful manipulation and the importance of rigorous verification.

Debunking the Myths: Addressing Incorrect Statements

In the spirit of thoroughness, it is essential to examine the incorrect statements presented alongside the original expression. By understanding why these statements are false, we reinforce our grasp of the simplification process and solidify our understanding of algebraic principles.

Let us revisit the incorrect statements:

A. The final simplified product is -3y^2 + 7y - 9.

This statement is demonstrably false. Our meticulous simplification process led us to the product -3y^2 + 37y - 84, a far cry from the expression presented in statement A. The discrepancy arises from a failure to correctly apply the distributive property and combine like terms. The coefficient of the 'y' term and the constant term are significantly different in the incorrect statement.

B. The final simplified product is -3y^2 + 37y - 6. (This statement is intentionally designed to be incorrect for illustrative purposes.)

This statement, too, falls short of the truth. While it shares the correct coefficient for the y^2 term and the y term, it stumbles on the constant term. Our careful calculations revealed the constant term to be -84, not -6. This error likely stems from a miscalculation during the combination of constant terms.

By dissecting these incorrect statements, we gain a deeper appreciation for the nuances of algebraic manipulation. We learn that even minor errors can lead to significant deviations from the correct answer. Meticulousness and attention to detail are paramount in the pursuit of mathematical accuracy.

Conclusion: A Triumph of Simplification

Our journey through the expression -3(y-5)^2-9+7y has been a testament to the power of methodical simplification and the importance of rigorous verification. We have successfully navigated the intricate web of exponents, parentheses, multiplication, and addition, arriving at the elegant simplified product: -3y^2 + 37y - 84. Along the way, we have debunked incorrect statements, reinforcing our understanding of algebraic principles and the pitfalls of careless manipulation.

This exercise serves as a reminder that even seemingly complex expressions can be tamed through a systematic approach. By adhering to the order of operations, employing the distributive property, and combining like terms, we can unravel the most intricate mathematical puzzles and reveal the underlying beauty of algebraic expressions. The final simplified product stands as a symbol of our triumph, a testament to the power of mathematical reasoning and the rewards of meticulous calculation.

In conclusion, consider the expression -3(y-5)^2-9+7y. After careful simplification and verification, we have confidently arrived at the final simplified product: -3y^2 + 37y - 84. This journey has not only provided us with a solution but also reinforced our understanding of algebraic principles and the importance of precision in mathematical manipulations.