Prove X^2(x-10)^2=1 If X = √((5+2√6)/(5-2√6))

Introduction

In this article, we delve into an intriguing mathematical problem that challenges our understanding of algebraic manipulation and radical simplification. The problem at hand involves demonstrating a specific equation given a defined value for x. Specifically, we are given that x is equal to the square root of the fraction (5 + 2√6) / (5 - 2√6), and our task is to prove that x^2(x - 10)^2 = 1. This problem is an excellent exercise in honing our skills in algebraic manipulation, rationalization, and equation-solving techniques. It requires a step-by-step approach, ensuring that each transformation is mathematically sound and justified. By working through this problem, we not only arrive at the solution but also reinforce our understanding of fundamental algebraic principles. We will explore the nuances of radical expressions, the importance of rationalizing denominators, and the power of strategic simplification in solving complex equations.

Problem Statement: If x = √((5+2√6)/(5-2√6)), show that x^2(x-10)2=1

The given equation presents a fascinating challenge. We are tasked with proving that if x is defined as the square root of the fraction (5 + 2√6) / (5 - 2√6), then the expression x^2(x - 10)^2 must equal 1. This problem is not immediately straightforward and requires a careful, methodical approach. To tackle this, we must first simplify the expression for x. The presence of the square root and the radical term in the fraction suggests that rationalizing the denominator is a crucial first step. Rationalizing the denominator involves eliminating the radical from the denominator of the fraction, which can be achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. Once we simplify x, we can then substitute this value into the expression x^2(x - 10)^2 and, through careful algebraic manipulation, demonstrate that it indeed equals 1. This exercise will test our ability to handle complex algebraic expressions, manipulate radicals, and apply fundamental algebraic identities. The solution will not only validate the equation but also enhance our problem-solving skills in algebra.

Step-by-Step Solution

1. Simplify the Expression for x

To begin, we are given that x = √((5 + 2√6) / (5 - 2√6)). The first crucial step in solving this problem is to simplify the expression for x. The complexity arises from the radical in the denominator, which we can eliminate by a process called rationalization. Rationalizing the denominator involves multiplying both the numerator and the denominator of the fraction by the conjugate of the denominator. The conjugate of (5 - 2√6) is (5 + 2√6). By multiplying both the numerator and the denominator by this conjugate, we can eliminate the square root from the denominator. This process is based on the algebraic identity (a - b)(a + b) = a^2 - b^2, which allows us to transform the denominator into a rational number. This initial simplification is critical as it sets the stage for subsequent steps, making the expression easier to handle and manipulate. The careful execution of this step ensures that we are working with a simplified form of x, which is essential for proving the final equation.

Multiplying both the numerator and denominator by the conjugate (5 + 2√6), we get:

x = √(((5 + 2√6)(5 + 2√6)) / ((5 - 2√6)(5 + 2√6)))

2. Expand and Simplify the Numerator and Denominator

Next, we need to expand and simplify both the numerator and the denominator of the expression. Expanding the numerator, (5 + 2√6)(5 + 2√6), involves multiplying each term in the first binomial by each term in the second binomial. This yields: 55 + 5(2√6) + (2√6)5 + (2√6)(2√6) = 25 + 10√6 + 10√6 + 46 = 25 + 20√6 + 24 = 49 + 20√6. Thus, the numerator simplifies to 49 + 20√6. For the denominator, we can use the difference of squares identity, (a - b)(a + b) = a^2 - b^2, where a = 5 and b = 2√6. This gives us: 5^2 - (2√6)^2 = 25 - 46 = 25 - 24 = 1. Therefore, the denominator simplifies to 1. This simplification is a direct consequence of multiplying by the conjugate, which eliminates the radical from the denominator. With the numerator and denominator simplified, the expression for x becomes more manageable, allowing us to proceed with further simplification and substitution in later steps. The careful expansion and simplification here are key to arriving at the correct value of x and ultimately proving the given equation.

Expanding the numerator:

(5 + 2√6)(5 + 2√6) = 25 + 10√6 + 10√6 + 24 = 49 + 20√6

Expanding the denominator:

(5 - 2√6)(5 + 2√6) = 25 - (2√6)^2 = 25 - 24 = 1

Therefore,

x = √(49 + 20√6)

3. Recognize the Perfect Square

After simplifying the expression under the square root, we have x = √(49 + 20√6). The next crucial step involves recognizing that the expression 49 + 20√6 is a perfect square. This recognition is key to further simplifying x and requires a keen eye for algebraic patterns. We can rewrite 49 + 20√6 as (a + b)^2 = a^2 + 2ab + b^2. By carefully comparing the terms, we look for values of a and b that satisfy this pattern. In this case, we can see that 49 can be considered as the sum of two squares, and 20√6 can be expressed as 2ab. Through some algebraic insight, we can identify that 49 + 20√6 is indeed the square of (5 + 2√6). Recognizing perfect squares is a powerful technique in simplifying radical expressions and is often a critical step in solving algebraic problems involving square roots. This step allows us to remove the square root, leading to a much simpler expression for x and paving the way for the subsequent steps in proving the given equation.

We need to recognize that 49 + 20√6 is a perfect square. We can rewrite it as:

49 + 20√6 = (a + b)^2 = a^2 + 2ab + b^2

Comparing terms, we can see that 49 + 20√6 = (5 + 2√6)^2

4. Simplify x

Having recognized that 49 + 20√6 is a perfect square, specifically (5 + 2√6)^2, we can now further simplify the expression for x. Recall that x = √(49 + 20√6). Since 49 + 20√6 is equal to (5 + 2√6)^2, we can substitute this into the expression for x. This gives us x = √((5 + 2√6)^2). The square root of a square is simply the original expression, provided it's non-negative. In this case, (5 + 2√6) is clearly positive, so we can safely say that x = 5 + 2√6. This simplification is a significant step forward as it eliminates the square root, giving us a concrete value for x. With this simplified value of x, we are now in a much better position to tackle the main equation and demonstrate that x^2(x - 10)^2 = 1. The simplified value of x makes the subsequent algebraic manipulations more straightforward and less prone to errors.

Thus,

x = √((5 + 2√6)^2) = 5 + 2√6

5. Substitute x into the Expression x^2(x-10)2

Now that we have a simplified expression for x, which is x = 5 + 2√6, we can proceed to the next crucial step: substituting this value into the expression x^2(x - 10)^2. This step involves replacing each instance of x in the expression with its equivalent value, 5 + 2√6. This substitution transforms the original expression into one involving only numerical values and radicals, setting the stage for further algebraic manipulation and simplification. The accurate substitution of x is critical as any error at this stage will propagate through the rest of the solution. Once we have made the substitution, we will then need to carefully expand and simplify the resulting expression, paying close attention to the order of operations and the rules of algebra. This substitution is a key step in bridging the gap between the given value of x and the equation we need to prove, making it a central part of the solution process.

Substituting x = 5 + 2√6 into x^2(x - 10)^2, we get:

(5 + 2√6)^2((5 + 2√6) - 10)^2

6. Simplify the Expression

After substituting x = 5 + 2√6 into the expression x^2(x - 10)^2, we obtain (5 + 2√6)^2((5 + 2√6) - 10)^2. The next step is to simplify this expression, which involves several algebraic manipulations. First, we simplify the term inside the second parentheses: (5 + 2√6) - 10 = -5 + 2√6. Now our expression looks like (5 + 2√6)^2(-5 + 2√6)^2. To proceed, we can recognize that (5 + 2√6)^2 and (-5 + 2√6)^2 are squares of binomials, which can be expanded using the formula (a + b)^2 = a^2 + 2ab + b^2. Alternatively, we can rewrite the expression as [(5 + 2√6)(-5 + 2√6)]^2, which simplifies the expansion process. Expanding the product inside the brackets gives us (5 + 2√6)(-5 + 2√6) = -25 + 10√6 - 10√6 + 4*6 = -25 + 24 = -1. Thus, the expression simplifies to (-1)^2, which is a significant reduction in complexity. This simplification highlights the importance of strategic algebraic manipulation and the recognition of patterns that can lead to easier calculations.

Simplifying the expression,

(5 + 2√6)^2((5 + 2√6) - 10)^2 = (5 + 2√6)^2(-5 + 2√6)^2

We can rewrite this as:

[(5 + 2√6)(-5 + 2√6)]^2

Expanding the product inside the brackets:

(5 + 2√6)(-5 + 2√6) = -25 + 10√6 - 10√6 + 24 = -1

7. Final Calculation

Having simplified the expression to [(5 + 2√6)(-5 + 2√6)]^2 and then further to (-1)^2, we are now at the final calculation step. This step is straightforward but crucial in reaching the conclusion. We simply need to evaluate (-1)^2, which means multiplying -1 by itself. The result of this operation is 1. Therefore, the entire expression x^2(x - 10)^2 simplifies to 1. This final calculation completes the proof, demonstrating that given x = 5 + 2√6, the equation x^2(x - 10)^2 = 1 holds true. This outcome validates the step-by-step approach we have taken, from simplifying the initial expression for x to substituting it into the main equation and then carefully simplifying the resulting terms. The final calculation not only provides the answer but also confirms the correctness of our algebraic manipulations and logical reasoning throughout the solution process.

Therefore,

(-1)^2 = 1

Conclusion

In conclusion, we have successfully demonstrated that if x = √((5 + 2√6) / (5 - 2√6)), then x^2(x - 10)^2 = 1. This proof involved a series of critical steps, each building upon the previous one. We began by simplifying the expression for x through rationalizing the denominator, which is a fundamental technique in handling radical expressions. This initial simplification was crucial as it transformed a complex expression into a more manageable form. We then recognized that the simplified expression under the square root was a perfect square, allowing us to further reduce x to a more straightforward value: 5 + 2√6. This recognition of algebraic patterns is a key skill in problem-solving. Next, we substituted this value of x into the equation x^2(x - 10)^2, which transformed the equation into a numerical expression involving radicals. The subsequent simplification involved expanding and combining terms, carefully following the order of operations. Through strategic algebraic manipulation, we reduced the expression to (-1)^2, which finally evaluated to 1, thus proving the given equation. This problem serves as an excellent illustration of how a methodical approach, combined with a solid understanding of algebraic principles, can lead to the solution of seemingly complex mathematical problems. The step-by-step process not only provides the answer but also enhances our understanding of algebraic techniques and problem-solving strategies.

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