Solving Algebra Equation \(x - \frac{1}{x} = 2\)
The problem asks us to find the value of the expression \(x^2 - \frac{1}{x^2}\) given the equation \(x - \frac{1}{x} = 2\). We are also told that \(x > 0\). Let's break down the steps to find the value.
Step-by-step Calculation
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Calculate \(x^2 + \frac{1}{x^2}\)
- We start with the given equation: \(x - \frac{1}{x} = 2\)
- To find terms involving squares, we can square both sides of the equation: \(\left(x - \frac{1}{x}\right)^2 = 2^2\)
- Using the algebraic identity \((a-b)^2 = a^2 - 2ab + b^2\), we expand the left side: \(x^2 - 2(x)\left(\frac{1}{x}\right) + \left(\frac{1}{x}\right)^2 = 4\)
- Simplify the expression. Note that \(2(x)\left(\frac{1}{x}\right)\) simplifies to \(2\): \(x^2 - 2 + \frac{1}{x^2} = 4\)
- Now, we can rearrange the equation to find the value of \(x^2 + \frac{1}{x^2}\): \(x^2 + \frac{1}{x^2} = 4 + 2\) \(x^2 + \frac{1}{x^2} = 6\)
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Evaluate the expression \(x^2 - \frac{1}{x^2}\)
- Recall the difference of squares formula: \(a^2 - b^2 = (a-b)(a+b)\). Applying this here with \(a=x\) and \(b=\frac{1}{x}\), we get: \(x^2 - \frac{1}{x^2} = \left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right)\)
- We are given the value \(x - \frac{1}{x} = 2\).
- We need to find the value of \(x + \frac{1}{x}\). We can find this using another algebraic identity related to squares: \((a+b)^2 = (a-b)^2 + 4ab\). Let \(a=x\) and \(b=\frac{1}{x}\): \(\left(x + \frac{1}{x}\right)^2 = \left(x - \frac{1}{x}\right)^2 + 4(x)\left(\frac{1}{x}\right)\)
- Substitute the known value \(x - \frac{1}{x} = 2\): \(\left(x + \frac{1}{x}\right)^2 = (2)^2 + 4\) \(\left(x + \frac{1}{x}\right)^2 = 4 + 4\) \(\left(x + \frac{1}{x}\right)^2 = 8\)
- To find \(x + \frac{1}{x}\), take the square root of both sides. Since we are given \(x > 0\), it means \(\frac{1}{x}\) is also positive. Therefore, their sum \(x + \frac{1}{x}\) must be positive: \(x + \frac{1}{x} = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}\)
- Now, substitute the values back into the difference of squares formula: \(x^2 - \frac{1}{x^2} = \left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right) = (2) (2\sqrt{2})\) \(x^2 - \frac{1}{x^2} = 4\sqrt{2}\)
Connecting Calculation to Answer Choice
Our detailed calculation shows that \(x^2 - \frac{1}{x^2}\) equals \(4\sqrt{2}\). However, reviewing the options, we see that the value \(6\) is provided as the correct answer.
In Step 1, we found that \(x^2 + \frac{1}{x^2} = 6\). This matches the value given in the options. It is possible the question intended to ask for \(x^2 + \frac{1}{x^2}\).
Based on this related calculation, the value \(6\) is obtained for \(x^2 + \frac{1}{x^2}\).
Similar Questions
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A group of 630 children is seated in rows for a group photo session. Each row contains three less children than the row in front of it. Which one of the following number of rows is not possible?
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What is the integer assigned to N?
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Who has the maximum amount of money?
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