If 4x2-x-4=0, Then Find The Value Of (x-1/x).
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Introduction
Mathematics often presents us with intriguing equations that challenge our understanding and problem-solving skills. One such equation is:
\[ 4x^2 - x - 4 = 0 \]
This quadratic equation serves as a foundation for exploring various algebraic manipulations and concepts. The primary goal here is to determine the value of the expression:
\[ x - \frac{1}{x} \]
This problem not only reinforces the application of quadratic equations but also introduces the idea of transforming expressions to simplify complex algebraic fractions. Whether you're a student preparing for exams, a teacher designing problem sets, or an enthusiast exploring algebraic relationships, understanding how to approach such questions is essential.
In this article, we will delve into the detailed steps to find the value of \( x - \frac{1}{x} \) given the quadratic equation, exploring multiple methods including substitution, algebraic manipulation, and the use of quadratic formulas. We will also discuss related concepts for a comprehensive understanding of the problem.
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Understanding the Given Quadratic Equation
The Quadratic Equation: \( 4x^2 - x - 4 = 0 \)
This is a standard quadratic form:
\[ ax^2 + bx + c = 0 \]
where:
- \( a = 4 \)
- \( b = -1 \)
- \( c = -4 \)
Our goal is to find the value of an expression involving \( x \), but first, we need to understand the nature of \( x \) by solving the quadratic.
Solving the Quadratic Equation
Quadratic equations can be solved using various methods:
- Factoring
- Completing the square
- Quadratic formula
Given the coefficients, the most straightforward method here is to apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
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Applying the Quadratic Formula
Let's substitute the known values:
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \times 4 \times (-4)}}{2 \times 4} \]
Simplify numerator:
\[ x = \frac{1 \pm \sqrt{1 - 4 \times 4 \times (-4)}}{8} \]
Calculate the discriminant:
\[ D = 1 - 4 \times 4 \times (-4) \]
\[ D = 1 - (-64) \]
\[ D = 1 + 64 = 65 \]
Therefore,
\[ x = \frac{1 \pm \sqrt{65}}{8} \]
The solutions are:
\[ x = \frac{1 + \sqrt{65}}{8} \quad \text{or} \quad x = \frac{1 - \sqrt{65}}{8} \]
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Finding the Value of \( x - \frac{1}{x} \)
Now that we know the solutions for \( x \), we can proceed to find the value of the expression:
\[ x - \frac{1}{x} \]
This expression involves both \( x \) and its reciprocal, which suggests that algebraic manipulation can lead us to a simplified form.
Approach 1: Direct Substitution
Using the solutions for \( x \), substitute into the expression:
\[ x - \frac{1}{x} \]
but this might be cumbersome as it involves radicals. Alternatively, an algebraic approach that does not depend on the specific roots is preferred.
Approach 2: Algebraic Manipulation
We can manipulate the given quadratic equation to relate to \( x - \frac{1}{x} \).
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Deriving \( x - \frac{1}{x} \) from the Quadratic Equation
Step 1: Express \( x + \frac{1}{x} \)
It is often easier to work with sums or differences involving \( x \) and \( \frac{1}{x} \). To find \( x - \frac{1}{x} \), we can consider the related expression:
\[ x + \frac{1}{x} \]
which can be linked to the quadratic equation.
Step 2: Multiply the original equation by \( \frac{1}{x} \)
Starting from:
\[ 4x^2 - x - 4 = 0 \]
Divide through by \( x \) (assuming \( x \neq 0 \)):
\[ 4x - 1 - \frac{4}{x} = 0 \]
Rearranged as:
\[ 4x - \frac{4}{x} = 1 \]
Divide both sides by 4:
\[ x - \frac{1}{x} = \frac{1}{4} \]
This is a pivotal step: we've directly expressed \( x - \frac{1}{x} \) in terms of a constant.
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Final Answer
From the above derivation, the value of \( x - \frac{1}{x} \) is:
\[ \boxed{\frac{1}{4}} \]
This remarkable result shows that regardless of which root \( x \) takes (since the derivation is based on the original quadratic), the value of \( x - \frac{1}{x} \) is \( \frac{1}{4} \).
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Additional Insights and Verification
Verifying the Result
To ensure the correctness of this conclusion, let's verify by substituting the roots back into the expression.
For \( x = \frac{1 + \sqrt{65}}{8} \):
Calculate \( x - \frac{1}{x} \):
- \( x = \frac{1 + \sqrt{65}}{8} \)
- \( \frac{1}{x} = \frac{8}{1 + \sqrt{65}} \)
Rationalize the denominator:
\[ \frac{8}{1 + \sqrt{65}} \times \frac{1 - \sqrt{65}}{1 - \sqrt{65}} = \frac{8(1 - \sqrt{65})}{(1)^2 - (\sqrt{65})^2} = \frac{8(1 - \sqrt{65})}{1 - 65} = \frac{8(1 - \sqrt{65})}{-64} \]
Simplify numerator:
\[ \frac{8(1 - \sqrt{65})}{-64} = -\frac{8(1 - \sqrt{65})}{64} = -\frac{1 - \sqrt{65}}{8} \]
Now, compute:
\[ x - \frac{1}{x} = \frac{1 + \sqrt{65}}{8} - \left( -\frac{1 - \sqrt{65}}{8} \right) = \frac{1 + \sqrt{65}}{8} + \frac{1 - \sqrt{65}}{8} = \frac{(1 + \sqrt{65}) + (1 - \sqrt{65})}{8} \]
Simplify numerator:
\[ 1 + \sqrt{65} + 1 - \sqrt{65} = 2 \]
Therefore:
\[ x - \frac{1}{x} = \frac{2}{8} = \frac{1}{4} \]
Similarly, for the other root, the value will also be \( \frac{1}{4} \), confirming the consistency.
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Conclusion
In this comprehensive exploration, we examined the quadratic equation:
\[ 4x^2 - x - 4 = 0 \]
and derived the solutions using the quadratic formula as:
\[ x = \frac{1 \pm \sqrt{65}}{8} \]
The key insight was to manipulate the original equation to directly find the value of the expression \( x - \frac{1}{x} \). By multiplying the original equation by \( \frac{1}{x} \) and simplifying, we found:
\[ x - \frac{1}{x} = \frac{1}{4} \]
This demonstrates the power of algebraic manipulation and substitution in solving equations involving complex expressions. The result is elegant and confirms that regardless of which root \( x \) represents, the value of \( x - \frac{1}{x} \) remains constant at \( \frac{1}{4} \).
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Related Concepts and Tips for Students
- Quadratic Equations: Always consider solving quadratic equations using the quadratic formula, especially when factoring isn't straightforward.
- Expression Manipulation: When dealing with expressions like \( x - \frac{1}{x} \), try to relate them to the original equation or to known identities.
- Rationalization: Rationalizing denominators is crucial when simplifying expressions involving radicals.
- Verification: Always verify your solutions by substituting back into the original expressions to confirm correctness.
- Common Mistakes: Be cautious with signs, especially when applying the quadratic formula and rationalizing denominators.
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Final Thoughts
Understanding how to manipulate quadratic equations and related expressions is fundamental in algebra. The ability to derive the value of complex expressions like \( x - \frac{1}{x} \) from given equations enhances problem-solving skills and deepens mathematical understanding. This approach not only applies to this specific problem but also prepares you for more advanced topics involving algebraic identities, equations, and functions.
Whether you're preparing
