Find the value of $k$ for which the given equation has real and equal roots.

$\left(k+1\right)x^2-2\left(k-1\right)x+1=0$

For what values of $k$, the roots of the equation $x^2+4x+k=0$ are real?

Let $f\left(x\right)=x^6-2x^5+x^3+x^2-x-1$ and $g\left(x\right)=x^4-x^3-x^2-1$ be two polynomials. Let $a,b,c$ and $d$ be the roots of $g\left(x\right)=0$. Then, the value of 

$f\left(a\right)+f\left(b\right)+f\left(c\right)+f\left(d\right)$ is

If  _____ , then the quadratic equation has no real roots.

The roots of the quadratic equation $a{x}^2+2bx+c=0, \left(a\ne 0\right)$ are real and unequal, then $b^2>$ _____

Suppose $\mathrm{A},\mathrm{B},\mathrm{C}$ are defined as

$\mathrm{A}=a^{2}b+a{b}^2-a^{2}c-a{c}^2$

$\mathrm{B}=b^{2}c+b{c}^2-a^{2}b-a{b}^2$

$\mathrm{C}=a^{2}c+a{c}^2-b^{2}c-b{c}^2$

Where $a>b>c>0$ and the equation $A{x}^2+Bx+C=0$ has equal roots, then $a,b,c$ are in