Find the sum of all integral values of \(c\) with \(c\le 25\) for which the equation \(y=x^2-7x-c\) has two rational roots.

Guest Jul 19, 2019

#1**+1 **

**Find the sum of all integral values of \(c\) with \(c\le 25\) for which the equation \(y=x^2-7x-c\) has two rational roots.**

\(\begin{array}{|lrcll|} \hline & x^2-7x-c &=& 0 \\ & x &=& \dfrac{7\pm \sqrt{49-4(-c)}}{2} \\\\ & x &=& \dfrac{7\pm \sqrt{49+4c}}{2} \\\\ \text{Two rational roots} & 49+4c &>& 0 \\ & 4c &>& -49 \\ & c &>& -\frac{49}{4} \\ & \mathbf{ c } &>& \mathbf{-12.25} \\ \hline \end{array}\)

\(\begin{array}{|lrcll|} \hline \text{Integral values} & c &=& \{ -12,\ -11,\ -10,\ \ldots \ ,\ -1,\ 0,\ 1,\ \ldots \ ,\ 25\} \\\\ \text{The sum of all integral values of $c$ with $c\le 25$} & &=& \left(\dfrac{-12+25}{2}\right)\cdot (25+13) \\ & &=& \dfrac{13\cdot 38}{2} \\ & &=& \mathbf{247} \\ \hline \end{array} \)

heureka Jul 19, 2019