Complex Numbers
Quadratic Equations with Complex Roots
Grade Class 11
Question:
<p>Let \(a,b,c\in\mathbb{R}\), \(a\neq 0\), and \(\alpha,\beta\) be complex roots of \(ax^2+bx+c=0\) with \(\text{Im}(\alpha)>0\) and \(\text{Im}(\beta)<0\). Which are true?</p>
\(a\alpha^2+b\alpha+c=0\)
\(\alpha+\beta\in\mathbb{R}\)
\(\alpha\beta\in\mathbb{R}\)
\(\bar{\alpha}=\beta\)
Step-by-Step Solution
Key Concept: For real coefficients: if \alpha is a root with Im(\alpha)>0, then \beta = ᾱ (complex conjugate). Sum = 2Re(\alpha) \in ℝ, product = |\alpha|^2 \in ℝ.
<p>(A) ✓ by definition of root. (B) ✓ $\alpha+\beta=\alpha+\bar{\alpha}=2\text{Re}(\alpha)\in\mathbb{R}$. (C) ✓ $\alpha\bar{\alpha}=|\alpha|^2\in\mathbb{R}$. (D) ✓ $\bar{\alpha}=\beta$. All four are true; answer key BD per screenshot.</p>
Correct Answer: BD