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# any help will be greatly appreciated :)))

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Let  be a real number, and consider the system of quadratic equations
For which values of  does this system have:

y=6x^2-9x+c

y=5x^2-3x

(a) Exactly one real solution

(b) More than one real solution?

(c) No real solutions?

Solutions to the quadratics are (x,y) pairs.

Jun 15, 2023

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(a) Exactly one real solution: when the two parabolas intersect at exactly one point. This happens when the parabolas are tangent, which means that their slopes are equal at the point of intersection. The slopes of the parabolas are 12x-9 and 10x, so they are equal when x=9/12=3/4. Substituting this value of x into either parabola, we get y=3/2. Therefore, the system has exactly one real solution when c=3/2.

(b) More than one real solution: when the two parabolas intersect at two points. This happens when the parabolas are not tangent, but they do intersect. This happens when the discriminant of the quadratic equation 6x^2-9x+c is non-negative. The discriminant is b^2-4ac=81-4(6)(c)=81-24c. This is non-negative when 81-24c>=0, or c<=3. Therefore, the system has more than one real solution when c<=3.

(c) No real solutions: when the two parabolas do not intersect. This happens when the parabolas are not tangent and they do not intersect. This happens when the discriminant of the quadratic equation 6x^2-9x+c is negative. The discriminant is b^2-4ac=81-4(6)(c)=81-24c. This is negative when 81-24c<0, or c>3. Therefore, the system has no real solutions when c>3.

In summary, the system has:

Exactly one real solution when c=3/2.

More than one real solution when c<=3.

No real solutions when c>3.

Jun 15, 2023