If x2 - 3x + 1 = 0, What is the value of \({x^{2\;}}

If x² - 3x + 1 = 0, What is the value of \({x^{2\;}} - \frac{1}{{{x^{2\;}}}}\) ?

A \(4\sqrt 3 \)

B \(3\sqrt 5 \)

C \(4\sqrt 5 \)

D \(2\sqrt 3 \)

Correct Answer: Option B

Solution:

আমাদেরকে বের করতে হবে, \({x^{2\;}} - \frac{1}{{{x^{2\;}}}}\) = \((x + \frac{1}{x})(x - \frac{1}{x})\) .......... (1)

অর্থাৎ আমরা \((x + \frac{1}{x})\) এবং \((x - \frac{1}{x})\) এর মান নিচের সমীকরণের সাহায্যে বের করবো ।

x² - 3x + 1 = 0

=> \(\frac{{{x^2}}}{x} - \frac{{3x}}{x} + \frac{1}{x} = \frac{0}{x}\) [x দ্বারা ভাগ করে ]

=> \(x - 3 + \frac{1}{x} = 0\)

=> \(x + \frac{1}{x} = 3\) [পক্ষান্তর করে ] ......... (2)

=> \({(x + \frac{1}{x})^{2\;}} = {(3)^{2\;}}\) [ বর্গ করে ]

=> \({(x - \frac{1}{x})^{2\;}} + 4 \times x \times \frac{1}{x} = 9\)

=> \({(x - \frac{1}{x})^{2\;}} + 4 = 9\)

=> \({(x - \frac{1}{x})^{2\;}} = 9 - 4\)

=> \({(x - \frac{1}{x})^{2\;}} = 5\)

\((x - \frac{1}{x}) = \sqrt 5 \) [বর্গমূল করে ] .......... (3)

অর্থাৎ আমরা পেলাম \((x + \frac{1}{x}) = 3\) এবং \((x - \frac{1}{x}) = \sqrt 5 \)

\({x^2} - \frac{1}{{{x^2}}} = (x + \frac{1}{x})(x - \frac{1}{x}) = 3 \times \sqrt 5 = 3\sqrt 5 \)