The problem asks us to find the value of $(x - \frac{1}{x})$ given that $(x + \frac{1}{x}) = 7$. We can use an algebraic identity relating these two expressions.
Algebraic Identity Application
Consider the squares of the two expressions:
- $(x + \frac{1}{x})^2 = x^2 + 2(x)(\frac{1}{x}) + \frac{1}{x^2} = x^2 + 2 + \frac{1}{x^2}$
- $(x - \frac{1}{x})^2 = x^2 - 2(x)(\frac{1}{x}) + \frac{1}{x^2} = x^2 - 2 + \frac{1}{x^2}$
By comparing these, we can derive the identity:
$(x - \frac{1}{x})^2 = (x + \frac{1}{x})^2 - 4$
Solving for $(x - \frac{1}{x})$
We are given $(x + \frac{1}{x}) = 7$. Substitute this value into the identity:
- Square the given value: $(x + \frac{1}{x})^2 = 7^2 = 49$
- Use the identity to find $(x - \frac{1}{x})^2$: $(x - \frac{1}{x})^2 = (x + \frac{1}{x})^2 - 4$ $(x - \frac{1}{x})^2 = 49 - 4$ $(x - \frac{1}{x})^2 = 45$
- Take the square root of both sides: $(x - \frac{1}{x}) = \pm\sqrt{45}$
- Simplify the radical $\sqrt{45}$: $\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$
Therefore, $(x - \frac{1}{x}) = \pm 3\sqrt{5}$. Since the options provided are positive, we select the positive value.
Final Answer
The value of $(x - \frac{1}{x})$ is $3\sqrt{5}$.
Similar Questions
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Important Questions from Identities
The coefficient of y in the expansion of (2y – 5) 3, is:
If x + y = 2 and \(\frac{1}{x}+\frac{1}{y}=\frac{18}{5}\) , then the value of (x 3+ y 3) is:
If x - y = 11 and \(\rm \frac{1}{x} - \frac{1}{y} = \frac{11}{24}\) then the value of x 3 - y 3 + x 2y 2 ?
If 2x 2- 8x - 1 = 0, then what is the value of \(\rm 8x^3 - \frac{1}{x^3}\) ?
If \(\rm x+ \frac{1}{x} = 4,\) then the value of \(\rm x^5 + \frac{1}{x^5}\) is:
