Q.18: Given $9x^2 + 25y^2 = 181$ And $xy = -6$, Find The Value Of $3x + 5y$. Q.19: Evaluate $29^3 - 11^3$ Using Identities. Q.20: Simplify The Expression: $(x^2 + Y^2 - Z^2)^2 - (x^2 - Y^2 + Z^2)^2$.

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This article delves into the fascinating world of algebraic identities and simplifications, focusing on solving complex expressions and equations. We will tackle three distinct problems, each requiring a unique approach and a solid understanding of algebraic principles. These problems not only test your ability to manipulate equations but also highlight the elegance and power of algebraic techniques. Through detailed explanations and step-by-step solutions, we aim to demystify these mathematical challenges and provide you with the tools to solve similar problems with confidence.

In this section, we address the problem of finding the value of 3x + 5y given the equations 9x^2 + 25y^2 = 181 and xy = -6. This problem combines quadratic expressions with a product relationship, requiring a strategic approach to isolate and determine the desired value. To solve this, we'll utilize the algebraic identity (a + b)^2 = a^2 + 2ab + b^2, adapting it to fit the given expressions. Our goal is to manipulate the given equations to form an expression that includes 3x + 5y, allowing us to solve for its value directly.

First, let's consider the expression (3x + 5y)^2. Expanding this using the identity mentioned above, we get:

(3x + 5y)^2 = (3x)^2 + 2(3x)(5y) + (5y)^2 = 9x^2 + 30xy + 25y^2

Notice that we already have the values for 9x^2 + 25y^2 and xy. We know that 9x^2 + 25y^2 = 181 and xy = -6. Substituting these values into the expanded equation, we get:

(3x + 5y)^2 = 181 + 30(-6) = 181 - 180 = 1

Now, we have (3x + 5y)^2 = 1. To find the value of 3x + 5y, we take the square root of both sides:

3x + 5y = ±√1 3x + 5y = ±1

Thus, the possible values for 3x + 5y are 1 and -1. To determine the correct value, we would typically need additional information or constraints. However, without further context, we can conclude that 3x + 5y can be either 1 or -1. This problem demonstrates the importance of recognizing algebraic patterns and strategically applying identities to simplify and solve equations. The ability to manipulate expressions and substitute known values is a crucial skill in algebra, allowing us to tackle complex problems with confidence and precision.

This section focuses on evaluating 29^3 - 11^3 using algebraic identities. Directly computing these cubes and subtracting them would be cumbersome and time-consuming. Instead, we will leverage the identity for the difference of cubes, which provides a more elegant and efficient solution. The relevant identity is:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Here, we can identify a as 29 and b as 11. Substituting these values into the identity, we get:

29^3 - 11^3 = (29 - 11)(29^2 + 29 * 11 + 11^2)

First, let's simplify the (29 - 11) term:

29 - 11 = 18

Now, we need to calculate the values inside the second parenthesis. We have:

29^2 = 841 29 * 11 = 319 11^2 = 121

Substituting these values back into the equation, we get:

29^3 - 11^3 = 18(841 + 319 + 121)

Now, let's add the numbers inside the parenthesis:

841 + 319 + 121 = 1281

Finally, we multiply this sum by 18:

18 * 1281 = 23058

Therefore, 29^3 - 11^3 = 23058. This problem effectively demonstrates the power of algebraic identities in simplifying complex calculations. By recognizing the pattern of the difference of cubes and applying the appropriate identity, we were able to avoid the tedious process of direct computation and arrive at the solution efficiently. The ability to identify and apply algebraic identities is a valuable skill in mathematics, allowing for streamlined problem-solving and a deeper understanding of algebraic relationships.

In this section, we tackle the simplification of the expression (x^2 + y^2 - z^2)^2 - (x^2 - y^2 + z^2)^2. This expression involves the difference of two squared terms, making it an ideal candidate for applying the difference of squares identity. The difference of squares identity is:

a^2 - b^2 = (a + b)(a - b)

Here, we can identify a as (x^2 + y^2 - z^2) and b as (x^2 - y^2 + z^2). Applying the identity, we get:

(x^2 + y^2 - z2)2 - (x^2 - y^2 + z2)2 = [(x^2 + y^2 - z^2) + (x^2 - y^2 + z2)][(x2 + y^2 - z^2) - (x^2 - y^2 + z^2)]

Now, let's simplify the expressions inside the brackets. For the first bracket, we have:

(x^2 + y^2 - z^2) + (x^2 - y^2 + z^2) = x^2 + y^2 - z^2 + x^2 - y^2 + z^2

Combining like terms, we get:

x^2 + x^2 + y^2 - y^2 - z^2 + z^2 = 2x^2

For the second bracket, we have:

(x^2 + y^2 - z^2) - (x^2 - y^2 + z^2) = x^2 + y^2 - z^2 - x^2 + y^2 - z^2

Combining like terms, we get:

x^2 - x^2 + y^2 + y^2 - z^2 - z^2 = 2y^2 - 2z^2

We can factor out a 2 from this expression:

2y^2 - 2z^2 = 2(y^2 - z^2)

Now, substituting these simplified expressions back into the equation, we get:

(x^2 + y^2 - z2)2 - (x^2 - y^2 + z2)2 = (2x2)[2(y2 - z^2)]

Multiplying the constants, we have:

= 4x^2(y^2 - z^2)

Finally, we can expand the expression further using the distributive property:

= 4x^2y^2 - 4x^2z^2

Thus, the simplified form of the given expression is 4x^2y^2 - 4x^2z^2. This problem showcases the effectiveness of the difference of squares identity in simplifying complex algebraic expressions. By recognizing the pattern and applying the identity, we were able to break down the expression into manageable parts and arrive at the simplified form. This technique is invaluable in algebra, allowing for efficient manipulation and simplification of expressions.

In this article, we have explored three distinct algebraic problems, each requiring the application of different algebraic principles and identities. We successfully found the value of 3x + 5y given quadratic and product relationships, evaluated 29^3 - 11^3 using the difference of cubes identity, and simplified a complex expression involving the difference of squares. These examples demonstrate the importance of a strong foundation in algebraic identities and techniques for effective problem-solving in mathematics. By mastering these concepts, you can approach complex problems with confidence and precision, unlocking the beauty and power of algebra.