The first equation in the system presented here represents the path of the sailboat. The second equation represents the path of the motorboat.

y = - 0.5 (x - 2) 2 + 0.5

y = 0.5 (x + 1) 2 - 1.5

You used substitution to obtain the equation 0 = x2 - x + 0.5 from the system. How many real solutions does the system have?

The system has no real solutions.

We use the discriminant to determine how many solutions it has:

b²-4ac

This is the value under the square root in the quadratic formula. If this value is negative, then there is no real solution; we cannot have a real number square root of a negative value.

If this value is 0, there is one real solution, and if this value is positive, there are 2 real solutions.

b²-4ac = (-1) ²-4 (1) (0.5) = 1-2=-1