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प्रश्न
If \[x + \frac{1}{x} = 12,\] find the value of \[x - \frac{1}{x} .\]
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उत्तर
Let us consider the following equation:
\[x + \frac{1}{x} = 12\]
Squaring both sides, we get:
\[\left( x + \frac{1}{x} \right)^2 = \left( 12 \right)^2 = 144\]
\[ \Rightarrow \left( x + \frac{1}{x} \right)^2 = 144\]
\[ \Rightarrow x^2 + 2 \times x \times \frac{1}{x} + \left( \frac{1}{x} \right)^2 = 144 [ (a + b )^2 = a^2 + b^2 + 2ab]\]
\[ \Rightarrow x^2 + 2 + \frac{1}{x^2} = 144\]
\[\Rightarrow x^2 + \frac{1}{x^2} = 142\] (Subtracting 2 from both sides)
Now
\[\left( x - \frac{1}{x} \right)^2 = x^2 - 2 \times x \times \frac{1}{x} + \left( \frac{1}{x} \right)^2 = x^2 - 2 + \frac{1}{x^2} [(a - b )^2 = a^2 + b^2 - 2ab]\]
\[ \Rightarrow \left( x - \frac{1}{x} \right)^2 = x^2 - 2 + \frac{1}{x^2}\]
\[ \Rightarrow \left( x - \frac{1}{x} \right)^2 = 142 - 2 ( \because x^2 + \frac{1}{x^2} = 142)\]
\[ \Rightarrow \left( x - \frac{1}{x} \right)^2 = 140 \]
\[ \Rightarrow x - \frac{1}{x} = \pm \sqrt{140} \left( \text { Taking square root } \right)\]
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Age (Human Years) |
| 2 | ||
| 3 | ||
| 4 | ||
| 5 | ||
| 6 |
