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19. If \( 2 x^{2}-6 x=1 \), then \( x^{2}+\frac{1}{4 x^{2}}= \) ?\( 2 x^{2}-6 x=1 \) అయితే \( x^{2}+\frac{1}{4 x^{2}}= \) విలువ ఎంత?1) 82) 123) 94) 10 |
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Answer» 2x2 - 6x = 1 ⇒ 2x2 - 6x - 1 = 0 ⇒ x = \(\frac{6\pm\sqrt{36+8}}4=\frac{6\pm2\sqrt{11}}4\) = \(\frac{3\pm \sqrt{11}}2\) ⇒ x2 = \(\frac{(3\pm\sqrt{11})^2}4\) = \(\frac{9+11\pm6\sqrt{11}}4\) = 5 \(\pm\frac32\sqrt{11}\) = \(\frac12\)(10 \(\pm\)3√11) Case I:- If x2 = \(\frac12\)(10 + 3√11) Then 1/4x2 = 1/4 x \(\frac2{10+3\sqrt{11}}\) = \(\frac12\times\frac{10-3\sqrt{11}}{100-99}\) = \(\frac{10-3\sqrt{11}}2\) ∴ x2 + 1/4x2 = \(\frac12\)((10+3√11) + (10 - 3√11)) = 20/2 = 10 Case II:- If x2 = \(\frac12\)(10 - 3√11) then 1/4x2 = 1/4 x \(\frac2{10-3√11}\) = \(\frac12\times\frac{10+3√11}{100-99}\) = \(\frac{10+3\sqrt{11}}2\) ∴ x2 + \(\frac1{4x^2}=10\) Hence, x2 + \(\frac1{4x^2}\) = 10 |
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