Verify that z 2 + 4i is a solution of z2 + 6z - 40i = 0. Is 2+ 41 also a solution?
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Transcribed Image Text:**Problem Statement:**
1. Verify that \( z = 2 + 4i \) is a solution of the equation \( z^2 + 6z - 40i = 0 \).
2. Is \( 2 + 4i \) also a solution?
3. Does your answer contradict Euler’s 1742 observation that solutions to polynomial equations with real coefficients come in conjugate pairs?
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**Explanation:**
This text queries the reader to verify a given complex number as a solution to a polynomial equation.
- **Complex Number Solution:** The equation \( z^2 + 6z - 40i = 0 \) suggests verifying whether the complex number \( z = 2 + 4i \) satisfies it.
- **Complex Conjugate:** The question raises whether the solution agrees with Euler's observation about solutions to polynomials having real coefficients, implying the expectation of complex conjugate pairs as solutions.
**Note:** The expression does not contain any graphs or diagrams, focusing entirely on mathematical analysis and algebraic verification.
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