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
If \(2x + { {1} \over 3x}= 5, x ≠ 0\) , then what is the value of \(27x^3+{{1} \over 8x^3}\) ?
If x 2 + 4y 2 = 40, xy = 6 and x > 2y then the value of x - 2y is:
- \(\dfrac{(0.73)^3+(0.31)^3}{(0.73)^2-0.73\times0.31+(0.31)^2}\)
- \(\dfrac{(5.17-2.19)^2-(5.17+2.19)^2}{11.3223}\)
If a(a + b + c) 2= 1792; b(a + b + c) 2= 1536; c(a + b + c) 2= 768 then what will be the value of a?
