Question

Solve for x ${(5+2\sqrt{6})}^{x^2-3}+{(5+2\sqrt{6})}^{x^2-3}=10$
product of the roots.

Answer 1

${(5+2\sqrt{6})}^{x^2-3}+{(5-2\sqrt{6})}^{x^2-3}=10$ ....(1)
Here $(5+2\sqrt{6})(5-2\sqrt{6})=1$
$\Rightarrow (5-2\sqrt{6})=\frac{1}{5+2\sqrt{6}}$
Using this in (1), we get
${(5+2\sqrt{6})}^{x^2-3}+{\left(\frac{1}{5+2\sqrt{6}}\right)}^{x^2-3}=10$
Put ${(5+2\sqrt{6})}^{x^2-3}=t$
$\Rightarrow t+\frac{1}{t}=10$
$\Rightarrow t^2-10t+1=0$
$\Rightarrow t=5\pm 2\sqrt{6}$
$\Rightarrow {(5+2\sqrt{6})}^{x^2-3}=5\pm 2\sqrt{6}$
$\Rightarrow x^2-3=1;x^2-3=-1$
$\Rightarrow x=\pm 2;\pm \sqrt{2}$
Product of roots =8