Answer
Diane: 9.1 hours, Lucy: 7.1 hours
Work Step by Step
Diane: 1/x completed in an hour
Lucy: 1/(x-2) completed in an hour
$1/x + 1/(x-2) = 1/4$
$1*x*(x-2)*4/x + 1*x*(x-2)*4/(x-2) = 1*x*(x-2)*4/4$
$(x-2)*4+4*x=x(x-2)$
$4x-8+4x=x^2-2x$
$8x-8=x^2-2x$
$x^2-10x+8=0$
$x=(-b±\sqrt {b^2-4ac})/2a$
$x=(-(-10)±\sqrt {(-10)^2-4*1*8})/2*1$
$x=(10±\sqrt {100-32})/2$
$x=(10±\sqrt {68})/2$
$x=(10±\sqrt {17*2*2})/2$
$x=(10±2\sqrt {17})/2$
$x=(5±\sqrt {17})$
$x=(5±\sqrt {17})$
$x=(5+\sqrt {17})$
$x=5+4.12$
$x=9.12$
$x=(5±\sqrt {17})$
$x=(5-\sqrt {17})$
$x=5-4.12$
$x=.88$ (not valid since Lucy would have a negative time)
