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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
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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