Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Our platform provides a seamless experience for finding precise answers from a network of experienced professionals. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
Given [tex]\( x = 3 - \sqrt{32} \)[/tex], we need to find the value of [tex]\( x^2 + \frac{1}{x^2} \)[/tex].
1. Calculate [tex]\( x^2 \)[/tex]:
[tex]\[ x = 3 - \sqrt{32} \] \[ x^2 = (3 - \sqrt{32})^2 \] \[ x^2 = 3^2 - 2 \cdot 3 \cdot \sqrt{32} + (\sqrt{32})^2 \] \[ x^2 = 9 - 6\sqrt{32} + 32 \] \[ x^2 = 41 - 6\sqrt{32} \][/tex]
2. Calculate [tex]\( \frac{1}{x} \)[/tex]:
[tex]\[ \frac{1}{x} = \frac{1}{3 - \sqrt{32}} \][/tex]
To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:
[tex]\[ \frac{1}{x} = \frac{1}{3 - \sqrt{32}} \cdot \frac{3 + \sqrt{32}}{3 + \sqrt{32}} \] \[ \frac{1}{x} = \frac{3 + \sqrt{32}}{(3 - \sqrt{32})(3 + \sqrt{32})} \] \[ \frac{1}{x} = \frac{3 + \sqrt{32}}{3^2 - (\sqrt{32})^2} \] \[ \frac{1}{x} = \frac{3 + \sqrt{32}}{9 - 32} \] \[ \frac{1}{x} = \frac{3 + \sqrt{32}}{-23} \] \[ \frac{1}{x} = -\frac{3 + \sqrt{32}}{23} \][/tex]
3. Calculate [tex]\( \left(\frac{1}{x}\right)^2 \)[/tex]:
[tex]\[ \left(\frac{1}{x}\right)^2 = \left(-\frac{3 + \sqrt{32}}{23}\right)^2 \] \[ \left(\frac{1}{x}\right)^2 = \frac{(3 + \sqrt{32})^2}{23^2} \] \[ \left(\frac{1}{x}\right)^2 = \frac{3^2 + 2 \cdot 3 \cdot \sqrt{32} + (\sqrt{32})^2}{529} \] \[ \left(\frac{1}{x}\right)^2 = \frac{9 + 6\sqrt{32} + 32}{529} \] \[ \left(\frac{1}{x}\right)^2 = \frac{41 + 6\sqrt{32}}{529} \][/tex]
4. Combine [tex]\( x^2 \)[/tex] and [tex]\( \left(\frac{1}{x}\right)^2 \)[/tex]:
[tex]\[ x^2 + \frac{1}{x^2} = (41 - 6\sqrt{32}) + \frac{41 + 6\sqrt{32}}{529} \] \[ x^2 + \frac{1}{x^2} = 41 - 6\sqrt{32} + \frac{41}{529} + \frac{6\sqrt{32}}{529} \][/tex]
Since the term ([tex]\( - 6\sqrt{32} \)[/tex] ) and ( [tex]+ 6\sqrt{32}[/tex] ) cancel each other out, we only need to add the constants:
[tex]\[ x^2 + \frac{1}{x^2} = 41 + \frac{41}{529} \] \[ x^2 + \frac{1}{x^2} = 41 + \frac{41}{529} \] \[ x^2 + \frac{1}{x^2} = 41 + 0.07751 \approx 41.0775 \][/tex]
So the value of [tex]\( x^2 + \frac{1}{x^2} \)[/tex] is approximately [tex]\( 41.0775 \)[/tex].
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
