Range and Graph of Quadratic Expressions

If $x^2+2ax+10-3a >0$ for all $x \in R$, then

If $a\in \left(m,n\right)$, then the sign of $x^2-(a+2)x+4$ is always positive. Find the value of $-m+n$?

Find the range of the following function:

$f\left(x\right)=\frac{1}{x-2}$.

Determine whether the graph of function is a parabola that opens upward or downward: $f\left(x\right)=6x^2+7x-9$.

Determine whether the graph of function is a parabola that opens upward or downward: $f\left(x\right)=-3x^2+2x-4$.

For real number $x$, if the minimum value of $f\left(x\right)=x^2+2bx+2c^2$ is greater than the maximum value of $g\left(x\right)=-x^2-2cx+b^2$, then

If $x^2+2\mathrm{px}-2\mathrm{p}+8>0$ for all real values of $x$ then the set of all possible values of $p$ is

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If $a,b$ and $c$ are real numbers such that $a^2+2b=7,b^2+4c=-7$ and $c^2+6a=-14,$ then find the value of$\frac{a^2+b^2+c^2}{2}$

If $x\in R$, then the difference between the maximum and minimum values of the expression $\frac{x^2+14x+9}{x^2+2x+3}$ is

Let $P\left(x\right)=4x^2+6x+7$ and $Q\left(y\right)=4y^2-12y+29.$ If $P\left(x\right).Q\left(y\right)=95$, then the value of $4(x-y)$ is

If graph of the equation $y=a{x}^2+bx+c$ is as below, then

Least integral value of $x$ satisfying the inequation $\left(x^2+1\right)<(x+2{)}^2<2x^2+4x-12$ is 

Vertex of the quadratic curve $y=a{x}^2+bx+c (a,b,c\in R)$ does not lie in the fourth quadrant, if :

The range of the function $f\left(x\right)=\frac{\left(2x^2+4x+7\right)}{\left(x^2+2x+4\right)}$ is $[a, b)$ then 

Let the minimum value of the real quadratic expression $a{x}^2-bx+\frac{1}{2a}$, $a>0$ be $y_0$. If $y_0$ occurs at $x=k$ and $k=2y_0$, then, the sum of possible value(s) of $b$ is $\lambda$. Then, the value of $\frac{\lambda +9}{5}$ is:

Let $p\left(x\right)=a{x}^2+bx+c$ be such that $p^2\left(0\right)+p^2\left(1\right)+p^2\left(2\right)+166=6 p\left(0\right)+12 p\left(1\right)+22 p\left(2\right),$ then

For given figure of $y=f\left(x\right)=a{x}^2+bx+c$,