If $x^2+3x-2k=0$ and $x^2-2x-7k=0$ have a non-zero common root,then the positive root of the equation $kx^2+(k+2)x-(k+1)=0$ is

$A$ value of $b$ for which the equations $x^2+bx-1=0$ and $x^2+x+b=0$ have one root in common is

If the two equations $x^2 - cx + d = 0$ and $x^2 - ax + b = 0$ have one common root and the second equation has equal roots,then $2(b + d) = $

Find the value of $a$ for which the equations $x^3+ax+1=0$ and $x^4+ax^2+1=0$ have a common root.

If $3x^2 - 7x + 2 = 0$ and $15x^2 - 11x + a = 0$ have a common root and $a$ is a positive real number,then the sum of the roots of the equation $15x^2 - ax + 7 = 0$ is:

Let $p_1(x) = x^3 - 2020x^2 + b_1x + c_1$ and $p_2(x) = x^3 - 2021x^2 + b_2x + c_2$ be polynomials having two common roots $\alpha$ and $\beta$. Suppose there exist polynomials $q_1(x)$ and $q_2(x)$ such that $p_1(x)q_1(x) + p_2(x)q_2(x) = x^2 - 3x + 2$. Then the correct identity is