We can analyze the system of quadratic equations to determine the number of real solutions for different values of c. Here's how:
Analyzing the Discriminant:
The discriminant of a quadratic equation determines the nature of its roots (solutions). It is denoted by the symbol b2−4ac. In this case, considering the first equation (y = 6x^2 - 9x + c):
a = 6
b = -9
c (variable)
The discriminant (d) for the first equation is:
d = (-9)^2 - 4 * 6 * c
The number of real solutions depends on the value of the discriminant:
d > 0: Two real and distinct solutions (roots)
d = 0: One repeated real solution (root)
d < 0: No real solutions (complex roots)
Relating Discriminant to c:
We want to find the values of c that correspond to each case.
(a) Exactly one real solution:
For exactly one real solution (repeated root), the discriminant needs to be zero.
Therefore, we need to solve:
0 = (-9)^2 - 4 * 6 * c
This simplifies to:
c = \frac{81}{24} = \dfrac{7}{2}
(b) More than one real solution:
For more than one real solution (distinct roots), the discriminant needs to be positive.
Therefore, we need to solve:
0 < (-9)^2 - 4 * 6 * c
This simplifies to:
c < \dfrac{81}{24} = \dfrac{7}{2}
(c) No real solutions:
For no real solutions (complex roots), the discriminant needs to be negative.
Therefore, we need to solve:
0 > (-9)^2 - 4 * 6 * c
This simplifies to:
c > \dfrac{81}{24} = \dfrac{7}{2}
Summary:
(a) Exactly one real solution: c = dfrac{7}{2}
(b) More than one real solution: c < dfrac{7}{2}
(c) No real solutions: c > dfrac{7}{2}
