To determine the true statement regarding the other solution to the quadratic function [tex]\( h \)[/tex] given that one of its solutions is [tex]\( -4 + 7i \)[/tex], we must consider the properties of quadratic equations with real coefficients.
Quadratic functions with real coefficients have solutions that come in conjugate pairs if the solutions are complex numbers. The conjugate of a complex number [tex]\( a + bi \)[/tex] is [tex]\( a - bi \)[/tex].
Given:
One solution is [tex]\( -4 + 7i \)[/tex].
Using the concept that the complex conjugate of this solution must also be a solution, we find that the other solution must be:
[tex]\[
-4 - 7i
\][/tex]
Now we match this with the given options:
A. Function [tex]\( h \)[/tex] has no other solutions.
- This is incorrect because quadratic functions of degree 2 always have two solutions.
B. The other solution to function [tex]\( h \)[/tex] is [tex]\( -4 - 77 \)[/tex].
- This is incorrect because the correct conjugate solution involves the imaginary part, not a real number alone.
C. The other solution to function [tex]\( h \)[/tex] is [tex]\( 4 - 7i \)[/tex].
- This is incorrect because it incorrectly changes the real part of the given solution.
D. The other solution to function [tex]\( h \)[/tex] is [tex]\( -4 - 7i \)[/tex].
- This is correct because it is the conjugate of the given solution [tex]\( -4 + 7i \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
