Question
समीकरण $e ^{4 x }+ e ^{3 x }-4 e ^{2 x }+ e ^{ x }+1=0$ के वास्तविक मूलों की संख्या है
Divide by e $2 x$
$\Rightarrow \quad \mathrm{e}^{2 \mathrm{x}}+\mathrm{e}^{\mathrm{x}}-4+\frac{1}{\mathrm{e}^{\mathrm{x}}}+\frac{1}{\mathrm{e}^{2 \mathrm{x}}}=0$
$\Rightarrow\left(\mathrm{e}^{2 \mathrm{x}}+\frac{1}{\mathrm{e}^{2 \mathrm{x}}}\right)+\left(\mathrm{e}^{\mathrm{x}}+\frac{1}{\mathrm{e}^{\mathrm{x}}}\right)-4=0$
$\Rightarrow\left(\mathrm{e}^{\mathrm{x}}+\frac{1}{\mathrm{e}^{\mathrm{x}}}\right)^{2}-2+\left(\mathrm{e}^{\mathrm{x}}+\frac{1}{\mathrm{e}^{\mathrm{x}}}\right)-4=0$
Let $\mathrm{e}^{\mathrm{x}}+\frac{1}{\mathrm{e}^{\mathrm{x}}}=\mathrm{t} \Rightarrow(\mathrm{e^x}-1)^{2}=0 \Rightarrow \mathrm{x}=0$
Number of real roots $=1$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.