Suppose that you are traveling straight from los angeles, california to denver, colorado, stopping overnight in page, arizona. the distance from l.a. to denver is 1,017 miles, and the road connecting the three cities is a straight line. if the distance from l.a. to page is 205 miles greater than the distance from page to denver, how far is l.a. from page, and how far is page from denver?

The answer is
LA to Page: 611 mi
Page to Denver: 406 mi

Total distance from LA to Denver is T.
The distance LA to Page is x
The distance Page to Denver is y.

T = x + y
T = 1017

x + y = 1017

The distance from L.A. to Page (x) is 205 miles greater than the distance from Page to Denver (y):
x = y + 205

The system of two equations:
x + y = 1017
x = y + 205

Substitute x from the second equation into the first one:
y + 205 + y = 1017
2y + 205 = 1017
2y = 1017 - 205
2y = 812
y = 812 / 2
y = 406 mi (Page to Denver)

x = y + 205 = 406 + 205 = 611 mi (LA to Page)

The answer is  LA to Page: 611 mi  and Page to Denver: 406 mi.

Yo need to use discriminant
for ax²+bx+c=0the discriminat is b²-4ac
if the discriminant is1. less than 0, then 0 real solutions2. equal to 0, then 1 real solutions3. if more than 0, then 2 real solutions
given
6x²-2x+7=0a=6b=-2c=7(-2)²-4(6)(7)4-168we can see this is less than 0no real solutions