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?

Guest Mar 8, 2017

#1**0 **

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

CPhill
Mar 8, 2017