Question

The number of real solutions of the equation $(15+4\sqrt{14}{)}^t+(15-4\sqrt{14}{)}^t=30$, where $t=x^2-2\mid x\mid$ are

• A. $0$
• B. $2$
• C. $4$
• D. $6$

Answer 1

The correct option is C $4$

For the equation, $(15-4\sqrt{14}{)}^t+(15+4\sqrt{14}{)}^t=30$ to be true,

$t=\pm 1$

Now, $t=x^2-2|x|$

$x^2-2|x|=\pm 1$

$x^2-2|x|+1=0$

$(|x|-1{)}^2=0$

$x=\pm 1$ and $x^2-2|x|-1=0$

$|x|=\frac{2\pm \sqrt{4+4}}{2}$

$|x|=1\pm \sqrt{2}$

Hence, $4$ values.