ON PLATO Select the correct answer.
Which function has exactly one real solution?

Attached Below

Option A: f(x) = –3x² + 30x – 75

Step-by-step 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.