To determine which quadratic function has exactly one real solution, we need to evaluate the discriminant of each quadratic equation. The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero. Let's evaluate the discriminants for the given functions:
1. For the function [tex]\( f(x) = -4x^2 + 9x \)[/tex]:
[tex]\[
a = -4, \quad b = 9, \quad c = 0
\][/tex]
[tex]\[
\Delta_A = 9^2 - 4(-4)(0) = 81
\][/tex]
Since [tex]\(\Delta_A = 81 \neq 0\)[/tex], the function [tex]\( f(x) = -4x^2 + 9x \)[/tex] does not have exactly one real solution.
2. For the function [tex]\( f(x) = 6x^2 + 11 \)[/tex]:
[tex]\[
a = 6, \quad b = 0, \quad c = 11
\][/tex]
[tex]\[
\Delta_B = 0^2 - 4(6)(11) = -264
\][/tex]
Since [tex]\(\Delta_B = -264 \neq 0\)[/tex], the function [tex]\( f(x) = 6x^2 + 11 \)[/tex] does not have exactly one real solution.
3. For the function [tex]\( f(x) = 2x^2 + 4x - 5 \)[/tex]:
[tex]\[
a = 2, \quad b = 4, \quad c = -5
\][/tex]
[tex]\[
\Delta_C = 4^2 - 4(2)(-5) = 16 + 40 = 56
\][/tex]
Since [tex]\(\Delta_C = 56 \neq 0\)[/tex], the function [tex]\( f(x) = 2x^2 + 4x - 5 \)[/tex] does not have exactly one real solution.
4. For the function [tex]\( f(x) = -3x^2 + 30x - 75 \)[/tex]:
[tex]\[
a = -3, \quad b = 30, \quad c = -75
\][/tex]
[tex]\[
\Delta_D = 30^2 - 4(-3)(-75) = 900 - 900 = 0
\][/tex]
Since [tex]\(\Delta_D = 0\)[/tex], the function [tex]\( f(x) = -3x^2 + 30x - 75 \)[/tex] has exactly one real solution.
Thus, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
