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Let c be a real number, and consider the system of quadratic equations

y = 6x^2 - 9x + c - x^2 - x

y = 5x^2 - 3x + x^2 + 2x


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?

 Dec 17, 2023
 #1
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(a) Exactly one real solution:

 

As before, for the system to have exactly one solution, the graphs of the quadratic equations must intersect precisely once. Subtracting the second equation from the first, we get:

 

y - (5x^2 - 3x + 7^2 + 11x) = 6x^2 - 9x + c - x^2 - 3x - 3x^2 + 8x

 

Simplifying, we obtain:

y - 5x^2 + 6x = c - 49

 

This again represents a linear equation in y and x. If c - 49 is non-zero, then the line defined by this equation will intersect the parabola of the second equation (5x^2 - 3x + 49 = y) at a single point, corresponding to a unique solution.

 

Therefore, for the system to have exactly one real solution, c must be any value except for 49.

 

(b) More than one real solution:

 

As before, multiple solutions can occur if the parabolas touch at multiple points:

 

Parabolas coincide completely: This is still impossible due to different constant terms.

 

Parabolas touch at multiple points: This happens when the difference equation (y - (5x^2 - 3x + 49)) collapses into a point. This occurs when c - 49 = 0, meaning c = 49. 

 

In this case, the difference equation becomes y - 5x^2 + 6x = 0, which represents a straight line that coincides with a part of the second parabola, leading to multiple solutions.

 

(c) No real solutions:

 

If the system has no real solutions, the parabolas should never intersect. This can happen when the line obtained from the difference equation (y - (5x^2 - 3x + 49)) never intersects the parabola of the second equation (5x^2 - 3x + 49 = y).

 

We found that when c = 49, the line coincides with a part of the second parabola, leading to multiple solutions. In all other cases (c ≠ 49), the line defined by the difference equation will always cut the second parabola at least once.

 

Therefore, the system never has no real solutions.

 

In conclusion:

 

(a) Exactly one real solution: c is any value except for 49.

(b) More than one real solution: c = 49.

(c) No real solutions: This doesn't occur for any value of c.

 Dec 17, 2023

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