Exercise ID208

Algebra → Exponential equations and inequalities → Exponential equations $$5^x-5^{3-x}=20$$ $$5^x-\frac{5^3}{5^x}=20$$ $$5^x=t\Rightarrow t-\frac{5^3}{t}=20$$ $$\overline{)t-\frac{125}{t}=20}·t$$ $$t^2-125=20t$$ $$t^2-20t-125=0$$ $$t^2-20t-125=0 \Rightarrow a=1 ; b=-20 ; c=-125$$ $$t_{1,2}=\frac{20\pm \sqrt{{\left(-20\right)}^2-4·1\left(-125\right)}}{2}=\frac{20\pm \sqrt{400+500}}{2}=\frac{20\pm \sqrt{900}}{2}=\frac{20\pm 30}{2}$$ $$t_1=25 ; t_2=-5$$ $$t_1=25 \Rightarrow 5^{x_1}=25 \Rightarrow 5^{x_1}=5^2 \Rightarrow x_1=2$$ $$t_2=-5 \Rightarrow 5^{x_2}>0 \Rightarrow 5^{x_2}\ne -5$$ $$x=2$$