Consider the quadratic equations \begin{align*} y &=3x^2 - 5x, \\ y &= 2x^2 - x - c \end{align*} where $c$ is a real constant. (a) For what value(s) of $c$ will the system have exactly one solution $(a, b)?$ (b) For what value(s) of $c$ will the system have more than one real solution? (c) For what value(s) of $c$ will the system have nonreal solutions?
+128407 y = 3x^2 - 5x
y = 2x^2 - x - c
Set these =
3x^2 - 5x = 2x^2 - x - c simplify
x^2 - 4x + c = 0
Using the discriminant.....the system will have one real solution whenever
4^2 - 4c = 0
4 - c = 0
4 = c
And the system will have more than one real solution whenever
4^2 - 4c > 0
4 - c > 0
4 > c → c < 4
And the system will have no real solutions whenever
c > 4
