In the equation kx^2 + 5x = 10k, find the other root if one root is -5.

Subtract 10k from both sides:[tex] kx^2+5x-10k = 0 [/tex]Assuming [tex]k\neq 0[/tex], divide both sides by k:[tex] x^2+\dfrac{5}{k}x-10 = 0[/tex]When you write a quadratic equation as [tex]x^2-sx+p [/tex], you know that the two solutions follow the properties[tex]x_1+x_2=s,\quad x_1x_2=p [/tex]So, in this case, we have[tex]x_1+x_2=-\dfrac{5}{k},\quad x_1x_2=-10 [/tex]Since we know that [tex]x_1=-5[/tex] we have:[tex]\begin{cases}-5+x_2=-\dfrac{5}{k}\\ -5x_2=-10\end{cases}[/tex]This system has solution [tex]k=\frac{5}{3},\ x=2[/tex]