Conditions for Common Roots

The equation $x^2-ax+b=0$ & $x^3-p{x}^2+qx=0$, where $b\ne 0, q\ne 0$, have one common root & the second equation has two equal roots. Prove that $2\left(q+b\right)=ap$.

If the quadratic equations $x^2+bx+ca=0$ & $x^2+cx+ab=0$ have a common root, prove that the equation containing their other roots is $x^2+ax+bc=0$

If the three equations $x^2+ax+12=0, x^2+bx+15=0$ and $x^2+\left(a+b\right)x+36=0$ have a common positive root, then value of $a+b$ is

If $x^2-11x+a$ and $x^2-14x+2a$ have a common factor, then $a$ is equal to