Range and Graph of Quadratic Expressions

The least value of the quadratic polynomial, $f\left(x\right)=\left(2p^2+1\right)x^2+2\left(4p^2-1\right)x+4\left(2p^2+1\right)$ for real values of $p$ and $x$ is

If $b^2-4ac>0$ then the graph of $y=a{x}^2+bx+c$

If $x$ is real then minimum value of $x^2-8x+17=0$ is

The equation $\sin x=x^2+x+1$ has-

If $f:R-\left\{2\right\}\to R$ is a function defined by $f\left(x\right)=\frac{x^2-4}{x-2},$ then range is

If $p>1$, then prove that the expression $\frac{x^2-2x+p^2}{x^2+2x+p^2}$ lies between $\frac{p-1}{p+1}$ and $\frac{p+1}{p-1}$ for all real values of $x$.

If $x$ is real, show that the expression $\frac{x^2+2x-11}{x-3}$ takes values which do not lie between $4$ and $12$.

Determine the range of the value of $x$ for which $\frac{x^2-3x+24}{x^2-3x+3}<4$.

If $x$ be real, show that the value of $\frac{2x^2-2x+4}{x^2-4x+3}$ cannot lie between $-7$ and $1$.

If $x$ be real, show that the value of $\frac{x^2+34x-71}{x^2+2x-7}$ cannot lie between $5$ and $9$.

If $x$ be real, show that $\frac{x^2-3x+4}{x^2+3x+4}$ lies between $7$ and $\frac{1}{7}$.

Show that the value of the expressio $\frac{\left(x-2\right)\left(x-4\right)}{\left(x-1\right)\left(x-3\right)}$ may be any real quantity, for real values of $x$.

Find the greatest value of $\frac{x^2-2x+4}{x^2+2x+4}$, for all real value of $x$.

If $x$ be real, find the greatest value of $\frac{x+2}{2x^2+3x+6}$

Find the real value of $x$ for which $8x-x^2-15$ is 

negative

For the real value of $x$ the  greatest value of   $8x-x^2-15$ is .

Show that the expression $(x^2-4x+7)$ is always positive for any real values of $x$.

Find the real values of $x$ for which $(x^2-4x+1)$ is always negative.

If $x$ is real, can the value of $3+2x-x^2$be greater than $4$?