Question : If $x^{2} - 7 x + 1 = 0$ and $0 < x < 1$ , what is the value of $x^{2} - \frac{1}{x^{2}} ?$
Option 1: $21 \sqrt{5}$
Option 2: $- 21 \sqrt{5}$
Option 3: $28 \sqrt{5}$
Option 4: $- 28 \sqrt{5}$

Question

Question : If $(x^{2} + \frac{1}{x^{2}}) = 7$ , and $0 < x < 1$ , find the value of $x^{2} - \frac{1}{x^{2}}$ .

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Answer 1

Correct Answer: $- 21 \sqrt{5}$

Solution : Given: $x^{2} - 7 x + 1 = 0$
⇒ $x + \frac{1}{x} = 7$
Squaring the equation, we get,
$(x + \frac{1}{x})^{2} = 7^{2}$
⇒ $x^{2} + \frac{1}{x^{2}} + 2 = 49$
⇒ $x^{2} + \frac{1}{x^{2}} = 47$
And we know, $(x - \frac{1}{x})^{2} = x^{2} + \frac{1}{x^{2}} - 2$
⇒ $(x - \frac{1}{x})^{2} = 47 - 2 = 45$
⇒ $x - \frac{1}{x} = \pm \sqrt{45}$
⇒ $x = - \sqrt{45}$ [As $0 < x < 1$ ]
Now consider, $x^{2} - \frac{1}{x^{2}}$
Using $a^{2} - b^{2} = (a - b) (a + b)$
$= (x + \frac{1}{x}) (x - \frac{1}{x})$
$= 7 \times (- \sqrt{45})$
$= - 7 \times 3 \sqrt{5} = - 21 \sqrt{5}$
Hence, the correct answer is $- 21 \sqrt{5}$ .

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Question : If x2−7x+1=0 and 0<x<1, what is the value of x2−1x2?Option 1: 215Option 2: −215Option 3: 285Option 4: −285

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Question : If $x^{2} - 7 x + 1 = 0$ and $0 < x < 1$ , what is the value of $x^{2} - \frac{1}{x^{2}} ?$
Option 1: $21 \sqrt{5}$
Option 2: $- 21 \sqrt{5}$
Option 3: $28 \sqrt{5}$
Option 4: $- 28 \sqrt{5}$