Simplifying $\frac{2(5)}{\frac{x \pm \sqrt{x^2-120}}{10}}$ A Step-by-Step Guide

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Introduction to the Expression

In the realm of mathematics, complex expressions often present themselves as intricate puzzles, challenging our understanding and analytical skills. The expression $\frac{2(5)}{\frac{x \pm \sqrt{x^2-120}}{10}}$ is one such example, a fascinating blend of arithmetic operations, algebraic variables, and a square root function, all interwoven into a multi-layered fraction. To truly grasp its essence, we must embark on a detailed journey, dissecting each component and exploring their interplay. This article serves as a comprehensive guide, meticulously walking through the simplification process, unraveling the underlying mathematical concepts, and highlighting potential applications.

The initial encounter with this expression might seem daunting, but fear not! By systematically applying the fundamental principles of mathematics, we can navigate its complexity and reveal its inherent structure. Our exploration will not only focus on arriving at a simplified form but also on understanding the domain of the variable x, the significance of the $\pm$ sign, and the behavior of the expression under various conditions. This holistic approach will equip you with a deeper understanding of not just this particular expression but also the broader mathematical landscape it inhabits.

Our journey begins with a step-by-step simplification of the expression. We will first address the numerator, then the denominator, and finally the entire fraction. Each step will be accompanied by a clear explanation, ensuring that the reasoning behind each manipulation is transparent. Furthermore, we will delve into the nuances of the square root term, paying close attention to the conditions under which it yields real numbers. This involves understanding the concept of the discriminant and its role in determining the nature of the roots of a quadratic equation. As we progress, we will encounter opportunities to apply various algebraic identities and techniques, solidifying your grasp of these essential tools.

This exploration is not merely an academic exercise; it is a gateway to appreciating the beauty and power of mathematical expressions. By understanding how to manipulate and interpret these expressions, we unlock the ability to model real-world phenomena, solve complex problems, and gain insights into the intricate workings of the universe. So, let us embark on this mathematical adventure together, with curiosity as our guide and a commitment to unraveling the mysteries of $\frac{2(5)}{\frac{x \pm \sqrt{x^2-120}}{10}}$.

Step-by-Step Simplification of the Expression

To effectively simplify the expression $\frac{2(5)}{\frac{x \pm \sqrt{x^2-120}}{10}}$, we will proceed systematically, addressing each part of the expression in a logical sequence. Our approach will involve simplifying the numerator, then the denominator, and finally combining the results to obtain the simplified form of the entire expression. This methodical approach ensures clarity and minimizes the risk of errors.

First, let's focus on the numerator. The numerator is simply the product of 2 and 5, which is a straightforward arithmetic operation. 2 multiplied by 5 equals 10. Therefore, the numerator simplifies to 10. This is our starting point, a simplified top half of the fraction that we can now use as we move on to the more complex denominator.

Now, we shift our attention to the denominator, which is $\frac{x \pm \sqrt{x^2-120}}{10}$. This part of the expression involves an algebraic variable x, a square root term, and the $\pm$ symbol, indicating two possible solutions. To simplify this, we first recognize that the entire expression is divided by 10. This means we have a fraction within a fraction, a scenario that can be simplified by understanding how division works in fractions.

The $\pm$ symbol is crucial as it represents two distinct possibilities: one where we add the square root term to x and another where we subtract it. This duality arises from the nature of square roots; both a positive and a negative root can satisfy the equation. We will need to keep this in mind as we proceed, as it implies that our final solution might also have two branches.

The square root term, $\sqrt{x^2-120}$, introduces a critical constraint on the possible values of x. The expression inside the square root, $x^2 - 120$, must be non-negative for the result to be a real number. This is because the square root of a negative number is not a real number; it enters the realm of complex numbers. Therefore, we have the inequality $x^2 - 120 \geq 0$, which we will need to solve to determine the valid range of values for x. This range is known as the domain of the expression, and it is a fundamental aspect of understanding the behavior of the expression.

Finally, to simplify the entire complex fraction, we recall the rule that dividing by a fraction is the same as multiplying by its reciprocal. This is a key step in untangling the nested fraction and bringing us closer to a simplified form. By applying this rule, we will transform the division by $\frac{x \pm \sqrt{x^2-120}}{10}$ into a multiplication by $\frac{10}{x \pm \sqrt{x^2-120}}$. This transformation will allow us to combine the numerator and denominator into a single, more manageable expression. The resulting expression will reveal the true nature of the original complex fraction and pave the way for further analysis and interpretation.

Inverting and Multiplying: Unveiling the Simplified Form

Having simplified the numerator and analyzed the denominator, we now arrive at the crucial step of combining these elements to obtain the simplified form of the expression. This involves applying the fundamental principle of dividing by a fraction, which is equivalent to multiplying by its reciprocal. This step is pivotal in untangling the complex fraction and revealing its underlying structure.

Recall that our expression is $\frac{2(5)}{\frac{x \pm \sqrt{x^2-120}}{10}}$. We have already simplified the numerator to 10. The denominator is $\frac{x \pm \sqrt{x^2-120}}{10}$. To divide 10 by this fraction, we multiply 10 by the reciprocal of the denominator. The reciprocal of $\frac{x \pm \sqrt{x^2-120}}{10}$ is $\frac{10}{x \pm \sqrt{x^2-120}}$.

Therefore, the expression becomes: 10 * $\frac{10}{x \pm \sqrt{x^2-120}}$. Multiplying the numerators, we get 10 * 10 = 100. The denominator remains $x \pm \sqrt{x^2-120}$. Thus, the expression simplifies to $\frac{100}{x \pm \sqrt{x^2-120}}$.

This simplified form is a significant step forward. It presents the expression as a single fraction, making it easier to analyze and manipulate. However, we can further refine this expression by addressing the square root in the denominator. A common technique for dealing with square roots in the denominator is to rationalize it. Rationalizing the denominator involves multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of $x + \sqrt{x^2-120}$ is $x - \sqrt{x^2-120}$, and the conjugate of $x - \sqrt{x^2-120}$ is $x + \sqrt{x^2-120}$.

Multiplying both the numerator and denominator by the conjugate will eliminate the square root from the denominator, making the expression even simpler and easier to work with. This process relies on the difference of squares identity: (a + b)(a - b) = a² - b². By applying this identity, we can transform the denominator into a form that no longer contains a square root. The choice of whether to use the plus or minus conjugate depends on the original $\pm$ sign in the denominator, ensuring that we are effectively eliminating the square root term.

This step of rationalizing the denominator is not just a cosmetic change; it often reveals hidden properties of the expression and makes it easier to analyze its behavior. For example, it can help in identifying singularities (points where the expression becomes undefined) or in evaluating the expression for specific values of x. The rationalized form also lends itself more readily to further algebraic manipulations, such as simplification or solving equations involving the expression. Therefore, rationalizing the denominator is a crucial technique in the arsenal of any mathematician, and its application here demonstrates its power in simplifying complex expressions.

Rationalizing the Denominator: A Key Simplification Technique

To further simplify the expression $\frac{100}{x \pm \sqrt{x^2-120}}$, we will employ the technique of rationalizing the denominator. This process eliminates the square root from the denominator, making the expression more manageable and revealing its underlying structure. Rationalization involves multiplying both the numerator and denominator by the conjugate of the denominator. This is a powerful algebraic manipulation that leverages the difference of squares identity to achieve its goal.

The conjugate of $x + \sqrt{x^2-120}$ is $x - \sqrt{x^2-120}$, and the conjugate of $x - \sqrt{x^2-120}$ is $x + \sqrt{x^2-120}$. We will consider both cases arising from the $\pm$ sign. Let's first consider the case where we have $x + \sqrt{x^2-120}$ in the denominator. We multiply both the numerator and denominator by its conjugate, $x - \sqrt{x^2-120}$:

$\frac{100}{x + \sqrt{x^2-120}} * \frac{x - \sqrt{x^2-120}}{x - \sqrt{x^2-120}}$

This gives us:

$\frac{100(x - \sqrt{x^2-120})}{(x + \sqrt{x^2-120})(x - \sqrt{x^2-120})}$

Applying the difference of squares identity, (a + b)(a - b) = a² - b², to the denominator, we get:

$\frac{100(x - \sqrt{x^2-120})}{x^2 - (x^2-120)}$

Simplifying the denominator:

$\frac{100(x - \sqrt{x^2-120})}{x^2 - x^2 + 120} = \frac{100(x - \sqrt{x^2-120})}{120}$

We can further simplify by dividing both the numerator and denominator by their greatest common divisor, which is 20:

$\frac{5(x - \sqrt{x^2-120})}{6}$

Now, let's consider the case where we have $x - \sqrt{x^2-120}$ in the denominator. We multiply both the numerator and denominator by its conjugate, $x + \sqrt{x^2-120}$:

$\frac{100}{x - \sqrt{x^2-120}} * \frac{x + \sqrt{x^2-120}}{x + \sqrt{x^2-120}}$

This gives us:

$\frac{100(x + \sqrt{x^2-120})}{(x - \sqrt{x^2-120})(x + \sqrt{x^2-120})}$

Again, applying the difference of squares identity to the denominator:

$\frac{100(x + \sqrt{x^2-120})}{x^2 - (x^2-120)}$

Simplifying the denominator:

$\frac{100(x + \sqrt{x^2-120})}{x^2 - x^2 + 120} = \frac{100(x + \sqrt{x^2-120})}{120}$

And simplifying by dividing both the numerator and denominator by 20:

$\frac{5(x + \sqrt{x^2-120})}{6}$

Thus, by rationalizing the denominator, we have obtained two simplified expressions, one for each case of the $\pm$ sign. These expressions are much easier to analyze and work with than the original complex fraction. This process highlights the power of algebraic manipulation in simplifying complex expressions and revealing their underlying structure.

Determining the Domain: Valid Values for x

As we've simplified the expression, it's crucial to determine the domain, which represents the set of all possible values of x for which the expression is defined and yields real numbers. The presence of the square root, $\sqrt{x^2-120}$, imposes a significant restriction on the domain. The expression inside the square root, $x^2 - 120$, must be greater than or equal to zero for the square root to be a real number. This constraint leads to an inequality that we must solve to find the valid range of x values.

The inequality is: $x^2 - 120 \geq 0$. To solve this inequality, we first add 120 to both sides: $x^2 \geq 120$. Now, we take the square root of both sides. Remember that when taking the square root of both sides of an inequality, we need to consider both the positive and negative roots:

$x \geq \sqrt{120}$ or $x \leq -\sqrt{120}$

We can simplify $\sqrt{120}$ by factoring out the largest perfect square. 120 can be factored as 4 * 30, so $\sqrt{120} = \sqrt{4 * 30} = 2\sqrt{30}$. Therefore, the domain is:

$x \geq 2\sqrt{30}$ or $x \leq -2\sqrt{30}$

This means that the expression is defined for all values of x that are greater than or equal to $2\sqrt{30}$ or less than or equal to $-2\sqrt{30}$. In interval notation, this can be written as:

$(-\infty, -2\sqrt{30}] \cup [2\sqrt{30}, \infty)$

It is essential to consider the domain when working with this expression. Substituting values of x outside this domain will result in imaginary numbers, which are not within the scope of real number analysis. Understanding the domain is not just a mathematical technicality; it is a fundamental aspect of interpreting the expression and its behavior. For instance, if we were using this expression to model a real-world phenomenon, the domain would represent the range of input values that are physically meaningful.

Furthermore, the domain can also provide insights into the symmetry and behavior of the expression. In this case, the domain is symmetric about the y-axis, indicating that the expression might exhibit some form of symmetry as well. This is a valuable piece of information that can guide further analysis and interpretation of the expression.

Simplified Expression and its Implications

Having meticulously simplified the expression $\frac{2(5)}{\frac{x \pm \sqrt{x^2-120}}{10}}$ and determined its domain, we now have a clearer understanding of its behavior and properties. The simplified form, obtained after rationalizing the denominator, is:

$\frac{5(x \mp \sqrt{x^2-120})}{6}$

Where the $\mp$ sign corresponds to the opposite sign of the original $\pm$ in the denominator. This means that if the original expression had a plus sign in the denominator, we now have a minus sign in the simplified form, and vice versa. This subtle but important change arises from the process of multiplying by the conjugate.

This simplified form offers several advantages. First, it eliminates the complex fraction, presenting the expression as a single fraction. Second, it removes the square root from the denominator, making it easier to analyze and manipulate. Third, it clearly shows the two possible outcomes arising from the $\pm$ sign, allowing us to consider each case separately.

The expression now consists of two distinct functions, one for the plus case and one for the minus case:

• For the plus case: $\frac{5(x - \sqrt{x^2-120})}{6}$
• For the minus case: $\frac{5(x + \sqrt{x^2-120})}{6}$

These two functions exhibit interesting relationships. They are conjugates of each other, differing only in the sign of the square root term. This conjugacy suggests a symmetry in their behavior, which can be further explored through graphical analysis or by examining their properties, such as their limits and derivatives.

Furthermore, the simplified form allows us to easily evaluate the expression for specific values of x within its domain. We simply substitute the value of x into the appropriate function (depending on the original $\pm$ sign) and perform the arithmetic operations. This is significantly easier than evaluating the original complex fraction, which required multiple steps and careful attention to order of operations.

The simplified expression also provides insights into the asymptotic behavior of the function. As x approaches positive or negative infinity, the square root term dominates the behavior of the expression. By analyzing the limits as x approaches infinity, we can gain a deeper understanding of how the function behaves for large values of x. This is a crucial aspect of understanding the function's overall characteristics and its potential applications.

In summary, the simplification process has not only made the expression more manageable but has also revealed its underlying structure and properties. The simplified form, along with the determined domain, provides a complete picture of the expression's behavior and allows for further analysis and interpretation.

Conclusion: The Power of Mathematical Simplification

In this comprehensive exploration, we have successfully unraveled the complexities of the expression $\frac{2(5)}{\frac{x \pm \sqrt{x^2-120}}{10}}$. We embarked on a step-by-step journey, simplifying the expression through meticulous algebraic manipulation, rationalizing the denominator, and carefully determining the domain. This process has highlighted the power of mathematical simplification in transforming seemingly intricate expressions into manageable and insightful forms.

We began by breaking down the expression into its constituent parts: the numerator and the denominator. We simplified the numerator through basic arithmetic and then focused our attention on the more complex denominator. We recognized the importance of the $\pm$ sign, which indicated two distinct possibilities, and the presence of the square root, which imposed a constraint on the domain of the expression.

Next, we applied the fundamental principle of dividing by a fraction, which is equivalent to multiplying by its reciprocal. This crucial step transformed the complex fraction into a simpler form, allowing us to proceed with further simplification. We then employed the technique of rationalizing the denominator, a powerful algebraic manipulation that eliminates square roots from the denominator and often reveals hidden properties of the expression.

Rationalizing the denominator involved multiplying both the numerator and denominator by the conjugate of the denominator. This process leveraged the difference of squares identity, a fundamental algebraic tool, to eliminate the square root term. The result was a simplified expression that was significantly easier to analyze and interpret.

Determining the domain was another critical step in our exploration. The presence of the square root imposed a restriction on the possible values of x, requiring us to solve an inequality. We found that the expression is defined for all values of x that are greater than or equal to $2\sqrt{30}$ or less than or equal to $-2\sqrt{30}$. This domain is essential for understanding the expression's behavior and for ensuring that we are working with real numbers.

Finally, we analyzed the simplified expression and discussed its implications. We observed that the simplified form clearly shows the two possible outcomes arising from the $\pm$ sign, and that the two resulting functions are conjugates of each other. We also discussed how the simplified form allows for easy evaluation of the expression for specific values of x and provides insights into its asymptotic behavior.

This exploration has demonstrated the importance of systematic simplification in mathematics. By breaking down complex expressions into smaller, more manageable parts, applying fundamental principles and techniques, and paying close attention to detail, we can unravel their mysteries and gain a deeper understanding of their behavior. The skills and concepts learned in this journey are applicable to a wide range of mathematical problems and will serve as a valuable foundation for further exploration in the world of mathematics. The power of mathematical simplification lies not just in arriving at a simpler form, but also in the insights and understanding gained along the way.