# Consider the system of quadratic equations \begin{align*} y &=3x^2 - 5x, \\ y &= 2x^2 - x - c, \end{align*}where $c$ is a real number. (a) For what value(s) of $c$ will the system have exactly one solution $(x,y)?$ (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 no real solutions? Solutions to the quadratics are $(x,y)$ pairs. Your answers will be in terms of $c,$ but make sure you address both $x$ and $y$ for each part.

Hello, we need to solve this system, c being a real number.

y=y, right? So, it comes.

We can compute the discriminant.

If the discriminant is 0, there is 1 solution.

It means for

And the solution is

If the discriminant is > 0, there are 2 real solutions.

It means 4(4-c) > 0 <=> 4-c > 0 <=>

And the solution are

If the discriminant is < 0, there are no real solutions.

It means 4(4-c) < 0 <=> 4-c < 0 <=>

There are no real solutions and the complex solutions are

Thank you.

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