Theory of Equations 1 Question 51
52. Solve for $x$ : $(5+2 \sqrt{6})^{x^{2}-3}+(5-2 \sqrt{6})^{x^{2}-3}=10(1985,5 M)$
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Solution:
- Given, $(5+2 \sqrt{6})^{x^{2}-3}+(5-2 \sqrt{6})^{x^{2}-3}=10$
Put $y=(5+2 \sqrt{6})^{x^{2}-3} \Rightarrow(5-2 \sqrt{6})^{x^{2}-3}=\frac{1}{y}$
From Eq. (i), $y+\frac{1}{y}=10$
$\Rightarrow \quad y^{2}-10 y+1=0 \Rightarrow y=5 \pm 2 \sqrt{6}$
$\Rightarrow \quad(5+2 \sqrt{6})^{x^{2}-3}=5+2 \sqrt{6}$
or $\quad(5+2 \sqrt{6})^{x^{2}-3}=5-2 \sqrt{6}$
$\Rightarrow \quad x^{2}-3=1$ or $x^{2}-3=-1$
$\Rightarrow \quad x= \pm 2$ or $x= \pm \sqrt{2}$
$\Rightarrow \quad x= \pm 2, \pm \sqrt{2}$
