Question 837796
We have *[tex \large \alpha + \beta = -\frac{5}{2}] and *[tex \large \alpha \beta = \frac{k}{2}], by Vieta's formulas.


Note that *[tex \large \alpha^2 + \beta^2 + \alpha \beta = (\alpha + \beta)^2 - \alpha \beta]. So *[tex \large \left(-\frac{5}{2} \right)^2 - \frac{k}{2} = \frac{21}{4}], or *[tex \large \frac{25}{4} - \frac{k}{2} = \frac{21}{4}]. This gives us k = 2.