y=c(x^2) + d
In the systems of equations above, c and d are constants . For which of the following values of c and does the system of equations have no real solutions?
A) c=-6, d=6
B) c=-5, d=4
C) c=6, d=4
D) c=6, d=5
Please help, thank you.
The system will have no real solutions when the graphs of the two equations don’t intersect. Easy one first: simplifies to , a horizontal line.
So we need to pick values for and that will prevent the parabola from intersecting the line.
Quick way to nail it is to remember that for a parabola in form:
- The leading coefficient tells you whether the parabola opens up (if it’s positive) or down (if it’s negative).
- The constant term tells you the -intercept.
So we need either a -intercept greater than 5 with a parabola that opens up, or a -intercept less than 5 with a parabola that opens down.
A) No, goes through (0, 6) and opens down.
B) YES! Opens down, has a -intercept of 4.
C) No, goes through (0, 4) and opens up.
D) No, goes through (0, 5) so obviously intersects the line.