Résoudre operatorname{og}_{2}(x^2+5x-2)=log_{2}(x^2+3x-6)+log_{4}9

$$\operatorname { og } _ { 2 } ( x ^ { 2 } + 5 x - 2 ) = \log _ { 2 } ( x ^ { 2 } + 3 x - 6 ) + \log _ { 4 } 9$$

Calculer g_2 (solution complexe)

$$\left\{\begin{matrix}g_{2}=\frac{\ln(x^{2}+3x-6)+\ln(3)}{\ln(2)o\left(x^{2}+5x-2\right)}\text{, }&x\neq \frac{\sqrt{33}-5}{2}\text{ and }x\neq \frac{-\sqrt{33}-5}{2}\text{ and }o\neq 0\text{ and }x\neq \frac{\sqrt{33}-3}{2}\text{ and }x\neq \frac{-\sqrt{33}-3}{2}\\g_{2}\in \mathrm{C}\text{, }&\left(x=-\frac{\sqrt{309}}{6}-\frac{3}{2}\text{ or }x=\frac{\sqrt{309}}{6}-\frac{3}{2}\right)\text{ and }o=0\end{matrix}\right,$$

Calculer o (solution complexe)

$$\left\{\begin{matrix}o=\frac{\ln(x^{2}+3x-6)+\ln(3)}{\ln(2)g_{2}\left(x^{2}+5x-2\right)}\text{, }&x\neq \frac{\sqrt{33}-5}{2}\text{ and }x\neq \frac{-\sqrt{33}-5}{2}\text{ and }g_{2}\neq 0\text{ and }x\neq \frac{\sqrt{33}-3}{2}\text{ and }x\neq \frac{-\sqrt{33}-3}{2}\\o\in \mathrm{C}\text{, }&\left(x=-\frac{\sqrt{309}}{6}-\frac{3}{2}\text{ or }x=\frac{\sqrt{309}}{6}-\frac{3}{2}\right)\text{ and }g_{2}=0\end{matrix}\right,$$

Calculer g_2

$$\left\{\begin{matrix}g_{2}=\frac{\ln(x^{2}+3x-6)+\ln(3)}{\ln(2)o\left(x^{2}+5x-2\right)}\text{, }&o\neq 0\text{ and }x\neq \frac{-\sqrt{33}-5}{2}\text{ and }\left(x<\frac{-\sqrt{33}-3}{2}\text{ or }x>\frac{\sqrt{33}-3}{2}\right)\\g_{2}\in \mathrm{R}\text{, }&\left(x=-\frac{\sqrt{309}}{6}-\frac{3}{2}\text{ or }x=\frac{\sqrt{309}}{6}-\frac{3}{2}\right)\text{ and }o=0\end{matrix}\right,$$

Calculer o

$$\left\{\begin{matrix}o=\frac{\ln(x^{2}+3x-6)+\ln(3)}{\ln(2)g_{2}\left(x^{2}+5x-2\right)}\text{, }&g_{2}\neq 0\text{ and }x\neq \frac{-\sqrt{33}-5}{2}\text{ and }\left(x<\frac{-\sqrt{33}-3}{2}\text{ or }x>\frac{\sqrt{33}-3}{2}\right)\\o\in \mathrm{R}\text{, }&\left(x=-\frac{\sqrt{309}}{6}-\frac{3}{2}\text{ or }x=\frac{\sqrt{309}}{6}-\frac{3}{2}\right)\text{ and }g_{2}=0\end{matrix}\right,$$