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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.

 Sep 4, 2023
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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!

 Sep 5, 2023

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