Let $c$ be a real number, and consider the system of quadratic equations \begin{align*} y &=6x^2 - 9x + c, \\ y &= 5x^2 - 3x. \end{align*}For which values of $c$ does this system have: (a) Exactly one real solution $(x,y)?$ (b) More than one real solution? (c) No real solutions? Solutions to the quadratics are $(x,y)$ pairs.
The system of equations can be written as
6x^2 - 9x + c - x^2 - x = 5x^2 - 3x 5x^2 - 10x + c = 0
The discriminant of this quadratic equation is
b^2 - 4ac = 100 - 4(5)(c) = 100 - 20c
(a) The system has exactly one real solution when the discriminant is 0.
100 - 20c = 0 => c = 5
(b) The system has more than one real solution when the discriminant is positive.
100 - 20c > 0 => c < 5
(c) The system has no real solution when the discriminant is negative.
100 - 20c < 0 => c > 5
Therefore, the system has:
Exactly one real solution when c = 5.
More than one real solution when 0 < c < 5.
No real solution when c > 5.
Here is a table summarizing the results:
c | Number of real solutions
---|---
0 < c < 5 | More than one
c = 5 | Exactly one
c > 5 | None
Hope this helps!
