If the two roots of the quadratic 3x2+5x+k are -5+i sqrt(11)/6, -5-i sqrt(11)/6, what is k?
+12528 If the two roots of the quadratic 3x2+5x+k are -5+i sqrt(11)/6, -5-i sqrt(11)/6, what is k?
+26367 If the two roots of the quadratic 3x2+5x+k are ( -5+i sqrt(11))/6, ( -5-i sqrt(11) )/6, what is k?
\(\text{Let $ x_1 = \dfrac{ -5+i \sqrt{11} }{6} $ } \\ \text{Let $ x_2 = \dfrac{ -5-i \sqrt{11} }{6} $ } \)
\(\begin{array}{|rcll|} \hline 3x^2+5x+k &=& 0 \quad & | \quad : 3 \\ x^2+\dfrac{5}{3}x+\underbrace{\dfrac{k}{3}}_{=x_1x_2} &=& 0 \\\\ \dfrac{k}{3} &=& x_1x_2 \\\\ \dfrac{k}{3} &=& \left( \dfrac{ -5+i \sqrt{11} }{6} \right) \left( \dfrac{ -5-i \sqrt{11} }{6} \right) \\\\ \dfrac{k}{3} &=& \dfrac{ (-5)^2-(i \sqrt{11})^2 }{36} \\\\ \dfrac{k}{3} &=& \dfrac{25-i^2 \cdot 11 }{36} \quad & | \quad i^2 = -1 \\\\ \dfrac{k}{3} &=& \dfrac{25-(-1) \cdot 11 }{36} \\\\ \dfrac{k}{3} &=& \dfrac{25+11 }{36} \\\\ \dfrac{k}{3} &=& \dfrac{36 }{36} \\\\ \dfrac{k}{3} &=& 1 \\\\ \mathbf{k} & \mathbf{=} & \mathbf{3} \\ \hline \end{array}\)
