Question

Solve the following equations.
$4sin3x+3=\sqrt{2sin3x+2}..$

Answer 1

$4sin3x+3=\sqrt{2sin3x+2}$

$\Rightarrow t=sin3x$

$\Rightarrow 4t+3=\sqrt{2t+2}$

$\Rightarrow 16t^2+9+24t=2t+2$

$\Rightarrow 16t^2+22t+7=0$

$\Rightarrow 16t^2+8t+14t+7=0$

$\Rightarrow 8t(2t+1)+7(2t+1)=0$

$t=-1/2$ or $t=-7/8$

$sin3x=-1/2$ or $sin3x=-7/8$

$3x=x\pi +(-1{)}^n(\pi /6)$ or $3x=x\pi +(-1{)}^{n}si{n}^{-1}(-7/8)$

$x=\frac{n\pi }{3}-(-1{)}^n\pi /18$ or $x=\frac{n\pi }{3}+(-1{)}^n\frac{si{n}^{-}1(-7/8)}{3}$