A lattice point in the xy-plane is a point both of whose coordinates are integers (not necessarily p

Consider the funstion (x)=${\displaystyle \sqrt{x-1}+10}$ for the domain ${\displaystyle [1,\mathrm{\infty })}$
Find ${\displaystyle f^{-1}\left(x\right)}$, where ${\displaystyle f^{-1}}$ is the inverse of f.
Also state the domain of ${\displaystyle f^{-1}}$ in interval notation

Find the two complex numbers whose sum is ${\displaystyle 5-i}$ and product is ${\displaystyle 8+i}$

Use the vertex ${\displaystyle (h,k)}$ and a point on the graph ${\displaystyle (x,y)}$ find the general form of the equation of the quadratic function.
${\displaystyle (h,k)=(1,0)(x,y)=(0,1)}$

Given two real numbers x, y so that ${\displaystyle x^2+y^2+xy+4=4y+3x}$. Prove that ${\displaystyle 3(x^3-y^3)+20x^2+2xy+5y^2+39x\le 100}$.

Graphing Quadratic Functions A quadratic function f is given. (a) Express f in standard form. (b) Find the vertex and xand y-intercepts of f. (c) Sketch a graph of f. (d) Find the domain and range of f.

Determining how the given number lies in relation to the roots of a quadratic equation
Say we have a quadratic ${\displaystyle x^2+px+q=0}$
If this quadratic has 2 roots, then:
For any number, ${\displaystyle \lambda }$, if ${\displaystyle {\lambda }^2+p\lambda +q<0}$ then ${\displaystyle \lambda }$ lies between both roots.
and for any number ${\displaystyle \lambda }$, if ${\displaystyle {\lambda }^2+p\lambda +q>0}$ then if ${\displaystyle 2\lambda +p>0}$, then ${\displaystyle \lambda <}$ either of the roots.
1) I don't understand graphical (physical) meaning - What is ${\displaystyle (2\lambda +p)}$ and what is ${\displaystyle 2\lambda +p<0\text{and}2\lambda +p>0}$
2) Also, what is ${\displaystyle {\lambda }^2+p\lambda +q>0}$ and ${\displaystyle {\lambda }^2+p\lambda +q<0}$.

${\displaystyle 15=4+3w²}$