Answer
Doug: $x=\frac{11+\sqrt {101}}{2}$, 10.5 hr
Karen: $x=\frac{9+\sqrt {101}}{2}$, 9.5 hr
Work Step by Step
Karen is 1 hour faster than Doug.
$\frac{1}{x-1} + \frac{1}{x} = \frac{1}{5}$
$5x*(x-1)*\frac{1}{x-1} + 5x*(x-1)*\frac{1}{x} = 5x*(x-1)*\frac{1}{5}$
$5x*1+5*(x-1)=x*(x-1)$
$5x+5x-5=x^2-x$
$10x-5=x^2-x$
$0=x^2-11x+5$
$x=(-b±\sqrt{b^2-4ac})/2a$
$x=(-(-11)±\sqrt{(-11)^2-4*1*5})/2*1$
$x=(11±\sqrt{121-20})/2$
$x=(11±\sqrt{101})/2$
$x=(11±10.02498)/2$
$x=(11±10.02498)/2$
$x=(11+10.02498)/2$
$x=21.02498/2$
$x=10.5$
$x=(11±10.02498)/2$
$x=(11-10.02498)/2$
$x=.99502/2$
$x=.4975$
Since this value of $x$ would make the first fraction negative, this value is not a solution.
$x=(11±10.02498)/2$
$x=(9±10.02498)/2$
$x=19.02498/2$
$x=9.5$
