I need help with variable expression

Question

Good day! I have this coirdinate equation: $$\frac{gt^2}{2}+{v_y}t-\frac{5}{3}R=0$$ $$v=\sqrt{\frac{10}{9}*gR}$$ How i can express variable $t$ from this equation? I calculated this as quadratic equation, and caught this: $$t=\sqrt{\frac{10R}{9g}}$$ But on site where i checked result, placed this expression: $$t=\sqrt{\frac{10}{3}*\frac{R}{g}}*(\frac{\sqrt{10}-1}{3})$$ How to catch this expression? Exlain me, please. Maybe, i have not enough mathematic skills.

Answer 1

this is an equation of the form $$ax^2+bx+c=0$$ the solutions for $x$ are given by $$x_{1,2}=-\frac{b}{2a}\pm\frac{1}{2a}\sqrt{b^2-4ac}$$ in our example $$t_1=1/3\,{\frac {-3\,v_{{y}}+\sqrt {30\,gR+9\,{v_{{y}}}^{2}}}{g}}$$ $$t_2=-1/3\,{\frac {3\,v_{{y}}+\sqrt {30\,gR+9\,{v_{{y}}}^{2}}}{g}}$$ solve this formula for $t$: $$1/2\,g{t}^{2}+1/3\,\sqrt {10}\sqrt {{\frac {R}{g}}}t-5/3\,R=0$$