Find the values of the real parameter $k$

Question

I have to find the values of the real parameter $k$ for which the equation:
$$x^2-7x+2k+3=0$$
has roots with different signs.
We need the condition $x_1x_2<0$ for the roots to be with different signs. Using Vieta's formulas: $x_1+x_2=-\frac{b}{a}=-\frac{-7}{1}=7$ and $x_1x_2=\frac{c}{a}=\frac{2k+3}{1}=2k+3.$ As I said $x_1x_2<0$, so $2k+3<0, k<-\frac{3}{2}$. Is this the correct solution and should I write the first Vieta's formula ($x_1+x_2=-\frac{b}{a}$)?

Answer 1

Demand $B^2>4AC$ and $C/A=2k+3<0$ for the roots of $Ax^2+Bx+C=0$, when $A,B,C$ are real. Here, you get $k<37/8$ and $k<-3/2$. The overlap of these two conditions is the answer: $$k<-3/2$$