Math Problem Solution
We are given the equation $x + \frac{1}{x} = 5$ and asked to find the value of the expression $\frac{2x}{x^2 - 1}$.
Step 1: Simplify the Target Expression
Divide the numerator and the denominator of the expression $\frac{2x}{x^2 - 1}$ by $x$. This is valid as $x=0$ does not satisfy $x + \frac{1}{x} = 5$.
$ \frac{2x}{x^2 - 1} = \frac{\frac{2x}{x}}{\frac{x^2 - 1}{x}} = \frac{2}{x - \frac{1}{x}} $Step 2: Find the Value of $x - \frac{1}{x}$
We know that $(x - \frac{1}{x})^2 = x^2 - 2(x)(\frac{1}{x}) + \frac{1}{x^2} = x^2 - 2 + \frac{1}{x^2}$.
First, find $x^2 + \frac{1}{x^2}$. Square the given equation $x + \frac{1}{x} = 5$:
$ \left(x + \frac{1}{x}\right)^2 = 5^2 $ $ x^2 + 2(x)\left(\frac{1}{x}\right) + \frac{1}{x^2} = 25 $ $ x^2 + 2 + \frac{1}{x^2} = 25 $ $ x^2 + \frac{1}{x^2} = 25 - 2 = 23 $Now substitute this value back into the expression for $(x - \frac{1}{x})^2$:
$ \left(x - \frac{1}{x}\right)^2 = \left(x^2 + \frac{1}{x^2}\right) - 2 = 23 - 2 = 21 $Therefore, $x - \frac{1}{x} = \pm \sqrt{21}$.
Step 3: Calculate the Final Value
Substitute the possible values of $x - \frac{1}{x}$ into the simplified expression from Step 1:
$ \frac{2}{x - \frac{1}{x}} $If $x - \frac{1}{x} = \sqrt{21}$, the value is $\frac{2}{\sqrt{21}}$.
If $x - \frac{1}{x} = -\sqrt{21}$, the value is $\frac{2}{-\sqrt{21}}$.
To match the options provided, we rationalize the first result:
$ \frac{2}{\sqrt{21}} = \frac{2 \times \sqrt{21}}{\sqrt{21} \times \sqrt{21}} = \frac{2\sqrt{21}}{21} $This result matches Option A.
Similar Questions
- If $X + Y = 2Z$ and $Z + W = 2X$, then:
If \(x + \frac{1}{x} = 2\sqrt{3}\), then find the value of \(x^{3} + \frac{1}{x^{3}}\).
Important Questions from Algebra
If 2x – y = 2 and xy = \(\frac{3}{2}\) , then what is the value of x 3– \(\frac{{{y^3}}}{8}\) ?
If (10a 3+ 4b 3) : (11a 3- 15b 3) = 7 : 5, then (3a + 5b) : (9a - 2b) =?
- If 4sin 2 θ = 3(1+ cos θ), 0° < θ < 90°, then what is the value of (2tan θ + 4sin θ - sec θ)?
The value of:
\(\frac{{\sin 23^\circ \cos 67^\circ + \sec52^\circ \sin38^\circ + \cos 23^\circ \sin 67^\circ + \rm cosec52^\circ \cos 38^\circ }}{{\rm cose{c^2}20^\circ - {{\tan }^2}70^\circ }}\)
If (x + y) 3+ 27(x - y) 3= (Ax - 2y)(Bx 2+ Cxy + 13y 2), then the value of A - B - C is:
