Question

The equation $3^{2x+1}-14-3^{x+1}=\sqrt{1+9^x+6\times 3^{x-1}}$ has

• A. 2 real roots
• B. 1 real and 1 extraneous root
• C. Only 1 real root
• D. No real root

Answer 1

The correct option is C Only 1 real root

Consider the equation.
$3^{2x+1}-14-3^{x+1}=\sqrt{1+9^x+6\times 3^{x-1}}$
$\Rightarrow (3^x{)}^2\times 3^1-14-3^x\times 3^1=\sqrt{1+(3^x{)}^2+6\times 3^x\times 3^{-1}}$
$\Rightarrow 3t^2-14-3t=\sqrt{1+t^2+2t}(Letting,3^x=t$
$\Rightarrow 3t^2-14-3t=1+t$
$\Rightarrow 3t^2-4t-15=0$
$\Rightarrow 3t^2-9t+5t-15=0$
$\Rightarrow 3t(t-3)+5(t-3)=0$
$\Rightarrow (t-3)(3t+5)=0$
$t=3ort=\frac{-5}{3}$
$\Rightarrow 3^x=3^{1}or{3}^x=\frac{-5}{3}$
$\Rightarrow x=1(Since,3^x\text{is always positive})$
Therefore, the given equation has only 1 real root.
Hence, the correct answer is option (3).