Answer:
Option A: f(x) = –3x² + 30x – 75
Explanation:
To know which option is correct, we shall determine the discriminant of each equation since it gives the nature of the root of the equation.
Discriminant = b² – 4ac
NOTE:
1. If b² – 4ac < 0, it means the equation has no real roots.
2. If b² – 4ac = 0, it means the equation has only one real root.
3. If b² – 4ac > 0, it means the equation has two real roots
This can be obtained as follow:
For Option A:
f(x) = –3x² + 30x – 75
a = –3
b = 30
c = –75
Discriminant = b² – 4ac
Discriminant = 30² – (4 × –3 × –75)
Discriminant = 900 – 900
Discriminant = 0
Thus, the discriminant is equal to zero. This implies that the equation has only one real root.
For Option B:
f(x) = 2x² + 4x – 5
a = 2
b = 4
c = –5
Discriminant = b² – 4ac
Discriminant = 4² – (4 × 2 × –5)
Discriminant = 16 – (–40)
Discriminant = 16 + 40
Discriminant = 56
Thus, the discriminant is greater than zero. This implies that the equation has two real roots.
For Option C:
f(x) = 6x² + 11
a = 6
b = 0
c = 11
Discriminant = b² – 4ac
Discriminant = 0² – (4 × 6 × 11)
Discriminant = 0 – 264
Discriminant = – 264
Thus, the discriminant is lesser than zero. This implies that the equation has no real roots.
For Option D:
f(x) = –4x² + 9x
a = –4
b = 9
c = 0
Discriminant = b² – 4ac
Discriminant = 9² – (4 × –4 × 0)
Discriminant = 81 – 0
Discriminant = 81
Thus, the discriminant is greater than zero. This implies that the equation has two real roots.
Summary:
Option A : Only one real root.
Option B : Two real roots.
Option C : No real roots.
Option D : Two real roots.
The correct answer is option A.