If 2+3i is one of the roots of 2x^3-9x^2+kx-13=0, k R, then the real root of this equation…

Question

If $2+3i$ is one of the roots of $2x^3-9x^2+kx-13=0$, $k\in\mathbb R$, then the real root of this equation:

Accepted Answer

Since coefficients are real, $2-3i$ is also a root. Product of all three roots $=\dfrac{13}2$; $(2+3i)(2-3i)=13$, so the real root $\gamma$ satisfies $13\gamma=\dfrac{13}2 \implies \gamma=\dfrac12.$

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