Evaluating Algebraic Expressions And Perfect Squares Nov 2015 Objectives

Understanding Perfect Squares and Evaluating Expressions

In this article, we will delve into a mathematics problem from November 2015, focusing on evaluating algebraic expressions and understanding the concept of perfect squares. Specifically, we will address the problem of finding the value of an expression involving variables k, p, and q, where k is derived from the condition that a quadratic expression is a perfect square. This exploration will not only reinforce fundamental algebraic principles but also showcase their practical application in problem-solving. Let's embark on this mathematical journey together, unraveling the intricacies of the problem and gaining a deeper understanding of algebraic manipulations and perfect square trinomials. By dissecting each step and providing clear explanations, we aim to equip you with the knowledge and skills necessary to tackle similar problems with confidence.

The cornerstone of this problem lies in the understanding of perfect square trinomials. A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. In other words, it's an expression of the form ax² + bx + c that can be written as (mx + n)² or (mx - n)², where a, b, c, m, and n are constants. Identifying and creating perfect square trinomials is a crucial skill in algebra, enabling us to simplify expressions, solve equations, and tackle a variety of mathematical challenges. The problem at hand specifically challenges us to find the value of k that transforms the expression x² - 6x + k into a perfect square. This requires us to delve into the relationship between the coefficients of the quadratic expression and the constants in its factored form. By carefully analyzing the structure of the expression and applying our knowledge of perfect squares, we can determine the precise value of k that satisfies the given condition. This process not only highlights the importance of pattern recognition in mathematics but also underscores the power of algebraic manipulation in solving problems.

To make x² - 6x + k a perfect square, we need to recall the pattern of a perfect square trinomial. Consider the binomial (x - a)². Expanding this, we get x² - 2ax + a². Comparing this with our expression x² - 6x + k, we can see that the coefficient of the x term (-6) corresponds to -2a. Therefore, -2a = -6, which implies a = 3. Now, the constant term k should correspond to a². Since a = 3, a² = 3² = 9. Thus, the value of k that makes x² - 6x + k a perfect square is 9. This crucial step forms the foundation for the subsequent evaluation of the expression involving k, p, and q. Without correctly determining the value of k, the final answer would be inaccurate. This emphasizes the sequential nature of mathematical problem-solving, where each step builds upon the previous one. The ability to recognize and apply the perfect square trinomial pattern is a fundamental algebraic skill that not only aids in solving this particular problem but also proves invaluable in various other mathematical contexts. Furthermore, the process of deriving the value of k reinforces the importance of understanding the relationships between the coefficients and constants in algebraic expressions.

Evaluating the Expression: (k² - p) / q

Now that we have determined the value of k to be 9, we can proceed to evaluate the expression (k² - p) / q, given that p = 5 and q = 1000. This involves substituting the known values of k, p, and q into the expression and performing the necessary arithmetic operations. This step is a practical application of algebraic substitution, a fundamental skill in mathematics that allows us to determine the value of an expression when the values of its variables are known. The process of substitution not only simplifies the expression but also allows us to arrive at a numerical answer, providing a concrete solution to the problem. By carefully substituting the values and following the order of operations, we can ensure the accuracy of our calculation and arrive at the correct final answer. This step underscores the importance of precision in mathematical calculations and the need to adhere to established rules and conventions.

Substituting the values, we get (9² - 5) / 1000. First, we calculate 9², which is 81. So the expression becomes (81 - 5) / 1000. Next, we subtract 5 from 81, which gives us 76. The expression now simplifies to 76 / 1000. This is a straightforward arithmetic calculation that follows the order of operations, ensuring that we perform the exponentiation before the subtraction and the subtraction before the division. The ability to accurately perform these basic arithmetic operations is crucial for success in algebra and other branches of mathematics. It is important to pay close attention to detail and double-check each step to minimize the risk of errors. By breaking down the calculation into smaller, manageable steps, we can ensure the accuracy of our results and build confidence in our mathematical abilities.

To leave the answer in standard form, we need to simplify the fraction 76 / 1000. Both 76 and 1000 are divisible by 4. Dividing both the numerator and the denominator by 4, we get 19 / 250. This is the simplified form of the fraction, and it represents the final answer in standard form. Simplifying fractions is an essential skill in mathematics, allowing us to express numbers in their most concise and easily understandable form. It also helps in comparing and manipulating different fractions more effectively. The process of simplification often involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. In this case, the GCD of 76 and 1000 is 4, which allows us to simplify the fraction to 19 / 250. This final step completes the evaluation of the expression and provides us with a clear and concise answer to the problem. The ability to simplify fractions and express them in standard form is a valuable skill that is widely applicable in various mathematical contexts.

Summary of the Solution

In summary, we first determined the value of k that makes x² - 6x + k a perfect square, which was found to be 9. We then substituted this value, along with p = 5 and q = 1000, into the expression (k² - p) / q. After performing the arithmetic operations, we simplified the resulting fraction to 19 / 250, which is the final answer in standard form. This problem encapsulates several key algebraic concepts, including perfect square trinomials, algebraic substitution, and fraction simplification. By mastering these concepts, we can confidently tackle a wide range of mathematical problems and develop a deeper understanding of algebraic principles. The ability to break down complex problems into smaller, manageable steps, as demonstrated in this solution, is a crucial skill for success in mathematics and beyond.

Key Takeaways

• Understanding and recognizing perfect square trinomials is fundamental to solving algebraic problems.
• Algebraic substitution allows us to evaluate expressions by replacing variables with their corresponding values.
• Simplifying fractions to their standard form is essential for clear and concise mathematical communication.
• Breaking down complex problems into smaller steps enhances accuracy and understanding.

By mastering these key takeaways, you can strengthen your mathematical foundation and improve your problem-solving skills.

Additional Practice Problems

To further solidify your understanding of these concepts, consider attempting the following practice problems:

1. Evaluate (m² + n) / r, where m is the value that makes x² + 8x + m a perfect square, n = 7, and r = 500.
2. Find the value of c that makes x² - 10x + c a perfect square, and then evaluate (c² - 3) / 200.
3. Determine the value of k if x² + 4x + k is a perfect square, and subsequently evaluate (k² + 1) / 100.

By working through these practice problems, you can reinforce your understanding of perfect squares, algebraic substitution, and fraction simplification, further enhancing your mathematical proficiency.

If...

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Once the complete question is provided, I will be able to analyze it, identify the relevant mathematical concepts and principles, and develop a step-by-step solution. This will not only address the specific problem at hand but also enhance your understanding of related mathematical topics. Remember, mathematics is a cumulative subject, and understanding each concept thoroughly is crucial for building a strong foundation and tackling more complex problems in the future.