Sure, let's go through the steps to solve the problem.
### Step-by-Step Solution
1. Given Information:
- One solution of the quadratic function [tex]\( f \)[/tex] is [tex]\( \sqrt{7} + 5i \)[/tex].
2. Properties of Quadratic Equations:
- Quadratic equations with real coefficients have their roots either as real numbers or as complex conjugate pairs when dealing with complex numbers.
- If one root of the quadratic equation is a complex number [tex]\( a + bi \)[/tex], then the other root must be its complex conjugate [tex]\( a - bi \)[/tex].
3. Applying the Property:
- In this case, the given root is [tex]\( \sqrt{7} + 5i \)[/tex].
- According to the property, the other root should be [tex]\( \sqrt{7} - 5i \)[/tex].
4. Verify Against Options:
- Option A: "Function [tex]\( f \)[/tex] has no other solutions." This is incorrect because a quadratic function must have two roots.
- Option B: "The other solution to function [tex]\( f \)[/tex] is [tex]\( \sqrt{7} - 5i \)[/tex]." This is correct based on the property of conjugate pairs.
- Option C: "The other solution to function [tex]\( f \)[/tex] is [tex]\( -\sqrt{7} + 5i \)[/tex]." This is incorrect.
- Option D: "The other solution to function [tex]\( f \)[/tex] is [tex]\( -\sqrt{7} - 5i \)[/tex]." This is also incorrect.
### Conclusion:
- The correct answer is B. The other solution to function [tex]\( f \)[/tex] is [tex]\( \sqrt{7} - 5i \)[/tex].
So, the answer is option B.
