Determining how the given number lies in relation to the roots of a quadratic equation

Question

Say we have a quadratic $x^2+px+q = 0$

If this quadratic has 2 roots, then:

For any number $\lambda$, if $\lambda^2+p\lambda+q < 0$ then $\lambda$ lies between both roots.

and

For any number $\lambda$, if $\lambda^2+p\lambda+q > 0$ then if $2\lambda+p>0$, then $\lambda > $ either of the roots, and if $2\lambda+p<0$, then $\lambda < $ either of the roots.

1) I don't understand graphical (physical) meaning - What is $(2\lambda+p)$ and what is $2\lambda+p<0$ and $2\lambda+p>0$

2) Also, what is $\lambda^2+p\lambda+q > 0$ and $\lambda^2+p\lambda+q < 0$

Answer 1

Since your equation has $2$ distinct real roots we can write $x^2+px+q=(x-\alpha)(x-\beta)$ with $\alpha>\beta$ and $\alpha+\beta = -p$

Case 1: Let be $\lambda$ such that $\lambda^2 + p\lambda + q<0$. If we use the second writing of the polynomial we have $(\lambda-\alpha)(\lambda-\beta)<0$ and it is possible if and only if $\beta < \lambda < \alpha$ (I'm using $\beta<\alpha$ here). This means that $\lambda$ lies between the two roots.

Case 2: Let be $\lambda$ such that $\lambda^2+p\lambda+q>0$. Consider $2$ subcases:

• If $2\lambda+p>0$ then you have: $\lambda > \frac{\alpha + \beta}{2}>\frac{\beta+\beta}{2}>\beta$. Using the second writing you have now $(\lambda-\alpha)(\lambda - \beta)>0$ and using the above inequality, the second factor is more than $0$. Then also the first factor has to be more than $0$. So $\beta < \alpha < \lambda$.
• If $2\lambda+p<0$ then you have: $\lambda < \frac{\alpha + \beta}{2}<\frac{\alpha+\alpha}{2}<\alpha$. Using the second writing you have now $(\lambda-\alpha)(\lambda - \beta)>0$ and using the above inequality, the first factor is less than $0$. Then also the second factor has to be less than $0$. So $\alpha>\beta>\lambda$.

The meaning of the condition $2\lambda +p>0$ is that $\lambda$ lies at the right of the $x$ coordinate of the vertex of the parabola. Simmetrically $2\lambda+p<0$ means that $\lambda$ lies at the left of the $x$ coordinate of the vertex of the parabola.