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Question Number 25226 by Mr eaay last updated on 06/Dec/17

Show that if x=3−(√3).Show that x^2 +((36)/x^2 )=24

$${Show}\:{that}\:{if}\:{x}=\mathrm{3}−\sqrt{\mathrm{3}}.{Show}\:{that}\:{x}^{\mathrm{2}} +\frac{\mathrm{36}}{{x}^{\mathrm{2}} }=\mathrm{24} \\ $$

Answered by naka3546 last updated on 06/Dec/17

x^2 = (3 − (√3))^2 = 12 − 6(√3) = 6 (2 − (√3)) x^2 + ((36)/x^2 ) = (12 − 6(√3)) + ((36)/((12 − 6(√3)))) = 6(2 − (√3)) + ((36)/(6(2 − (√3)))) = 6(2 − (√3)) + (6/((2 − (√3)))) = 6(2 − (√3)) +( (6/((2 − (√3)))) × ((2 + (√3))/(2 + (√3))) ) = 6(2 − (√3)) +( ((6(2 + (√3)))/(4 − 3)) ) = 6 (2 − (√3) ) + 6 (2 + (√3)) = 6 (2 − (√3) + 2 + (√3)) = 6 × 4 = 24 x^2 + ((36)/x^2 ) = 24 (proved) .

$${x}^{\mathrm{2}} \:\:=\:\:\left(\mathrm{3}\:−\:\sqrt{\mathrm{3}}\right)^{\mathrm{2}} \:\:=\:\:\mathrm{12}\:−\:\mathrm{6}\sqrt{\mathrm{3}}\:\:=\:\:\mathrm{6}\:\left(\mathrm{2}\:−\:\sqrt{\mathrm{3}}\right) \\ $$$$ \\ $$$${x}^{\mathrm{2}} \:+\:\frac{\mathrm{36}}{{x}^{\mathrm{2}} } \\ $$$$=\:\:\left(\mathrm{12}\:−\:\mathrm{6}\sqrt{\mathrm{3}}\right)\:+\:\frac{\mathrm{36}}{\left(\mathrm{12}\:−\:\mathrm{6}\sqrt{\mathrm{3}}\right)} \\ $$$$=\:\:\mathrm{6}\left(\mathrm{2}\:−\:\sqrt{\mathrm{3}}\right)\:+\:\frac{\mathrm{36}}{\mathrm{6}\left(\mathrm{2}\:−\:\sqrt{\mathrm{3}}\right)} \\ $$$$=\:\:\mathrm{6}\left(\mathrm{2}\:−\:\sqrt{\mathrm{3}}\right)\:+\:\frac{\mathrm{6}}{\left(\mathrm{2}\:−\:\sqrt{\mathrm{3}}\right)} \\ $$$$=\:\:\mathrm{6}\left(\mathrm{2}\:−\:\sqrt{\mathrm{3}}\right)\:+\left(\:\frac{\mathrm{6}}{\left(\mathrm{2}\:−\:\sqrt{\mathrm{3}}\right)}\:×\:\frac{\mathrm{2}\:+\:\sqrt{\mathrm{3}}}{\mathrm{2}\:+\:\sqrt{\mathrm{3}}}\:\right) \\ $$$$=\:\:\mathrm{6}\left(\mathrm{2}\:−\:\sqrt{\mathrm{3}}\right)\:+\left(\:\frac{\mathrm{6}\left(\mathrm{2}\:+\:\sqrt{\mathrm{3}}\right)}{\mathrm{4}\:−\:\mathrm{3}}\:\right) \\ $$$$=\:\:\mathrm{6}\:\left(\mathrm{2}\:−\:\sqrt{\mathrm{3}}\:\right)\:+\:\mathrm{6}\:\left(\mathrm{2}\:+\:\sqrt{\mathrm{3}}\right) \\ $$$$=\:\:\mathrm{6}\:\left(\mathrm{2}\:−\:\sqrt{\mathrm{3}}\:+\:\mathrm{2}\:+\:\sqrt{\mathrm{3}}\right) \\ $$$$=\:\:\mathrm{6}\:×\:\mathrm{4} \\ $$$$=\:\:\mathrm{24} \\ $$$$ \\ $$$${x}^{\mathrm{2}} \:+\:\frac{\mathrm{36}}{{x}^{\mathrm{2}} }\:\:=\:\:\mathrm{24}\:\:\:\left({proved}\right)\:. \\ $$

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