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The quadratic equation given is:

\[
x^2 + bx + c = 0
\]

It has solutions \(-7 + 4i\) and \(-7 - 4i\). These solutions are complex conjugates, meaning the sum and product of the roots will help us determine the coefficients \(b\) and \(c\).

### Using Vieta's formulas:
For a quadratic equation \(x^2 + bx + c = 0\) with roots \(r_1\) and \(r_2\):
1. The sum of the roots is \(-b\).
2. The product of the roots is \(c\).

In this case:
- The sum of the roots \( (-7 + 4i) + (-7 - 4i) = -14 \).
  So, \(-b = -14\), which gives \(b = 14\).
  
- The product of the roots \(( -7 + 4i)( -7 - 4i) = (-7)^2 - (4i)^2 = 49 - (-16) = 49 + 16 = 65\).
  So, \(c = 65\).

### Therefore:
- \(b = 14\)
- \(c = 65\)

Would you like more details or clarifications?

Here are 5 related questions to explore this further:
1. What happens if the roots are not complex conjugates?
2. How do you derive the quadratic formula from Vieta's relations?
3. Can you use another method to find \(b\) and \(c\) from the given roots?
4. How are the coefficients related to the nature of the roots (real vs. complex)?
5. Can Vieta’s formulas be used for higher-degree polynomials?

**Tip:** For quadratic equations with complex conjugate roots, the imaginary parts always cancel out in the sum of the roots, leaving only the real part to determine \(b\).

Suppose the quadratic equation x^2 + bx + c = 0 has the solutions -7 + 4i and -7 - 4i. What are b and c?

This problem involves finding the coefficients b and c of a quadratic equation with complex roots -7 + 4i and -7 - 4i. Using Vieta's formulas, the sum and product of the roots help determine the values of b and c. Learn how to solve this algebra problem step by step.

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