Question

If $9^x+{12}^x={16}^x,$ then the equation has

• A. no real solution
• B. only one real solution
• C. two real solutions
• D. three real solutions

Answer 1

The correct option is B only one real solution

$9^x+{12}^x={16}^x$
$\Rightarrow 1+\frac{{12}^x}{9^x}=\frac{{16}^x}{9^x}$
$\Rightarrow 1+{\left(\frac{4}{3}\right)}^x={\left(\frac{4}{3}\right)}^{2x}$

Let ${\left(\frac{4}{3}\right)}^x=t$
Then $1+t=t^2$
$\Rightarrow t=\frac{1+\sqrt{1+4}}{2}=\frac{1+\sqrt{5}}{2}[\because t>0]$