Question

Solve the following equation for x ${(15+4\sqrt{14})}^t+{(15-4\sqrt{14})}^t=30$ where $t=x^2-2\left|x\right|$
find the number of roots of the equation.

Answer 1

${(15+4\sqrt{14})}^t+{(15-4\sqrt{14})}^t=30$ ....(1)
Here, $(15+4\sqrt{14})(15-4\sqrt{14})=1$
$(15-4\sqrt{14})=\frac{1}{(15+4\sqrt{14})}$
Using this in eqn (1), we get
${(15+4\sqrt{14})}^t+{\left(\frac{1}{15+4\sqrt{14}}\right)}^t=30$
Put ${(15+4\sqrt{14})}^t=u$
$\Rightarrow u^2-30u+1=0$
$\Rightarrow u=15\pm 4\sqrt{14}$
$\Rightarrow {(15+4\sqrt{14})}^t=15\pm 4\sqrt{14}$
$\Rightarrow t=1,-1$
$\Rightarrow x^2-2|x|=1;x^2-2|x|=-1$
Now, if $x\ge 0$
$\Rightarrow x^2-2x-1=0$ ; $x^2-2x+1=0$
$\Rightarrow x=1\pm \sqrt{2}$ ; $x=1$
$\therefore x=1+\sqrt{2};x=1$
Now, if $x<0$
$x^2+2x-1=0;x^2+2x+1=0$
$\Rightarrow x=-1\pm \sqrt{2}$ ; $x=-1$
$\therefore x=-1-\sqrt{2};x=-1$
So, the number of roots of the eqn are 4