The quadratic equations $x^2 - 6x + a = 0$ and $x^2 - cx + 6 = 0$ have a common root. The other roots of the first and second equations are integers in the ratio $4:3$. Find the common root.

The equations $5x^2 + 12x + 13 = 0$ and $ax^2 + bx + c = 0$ have a common root,where $a, b, c$ are the sides of $\Delta ABC$. Find $\angle C$ in degrees.

For $a \neq b$,if the equations $x^2+ax+b=0$ and $x^2+bx+a=0$ have a common root,then the value of $a+b=$

Statement-$I$: If $a, b, c \in R$ and the equations $ax^2 + bx + c = 0$ and $x^2 + 3x + 4 = 0$ have a common root,then $\frac{a+c}{b} = \frac{4}{3}$.
Statement-$II$: If $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$ have both roots common,then $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$,where $a_1, a_2, b_1, b_2, c_1, c_2 \in R$.

Let $E_1 \equiv ax^2+bx+c$,$E_2 \equiv bx^2+cx+a$,$E_3 \equiv cx^2+bx+a$ and $\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}=3$. If these quadratic expressions have a common zero,then the quadratic expression having zeroes that are common to $E_2$ and $E_3$ and different from the zeroes of $E_1$ is

Let $a \ne b, c \ne 0$. If the equations $x^2 + ax + bc = 0$ and $x^2 + bx + ac = 0$ have a common root,then:
Statement-$1$: The equation of the other roots is $x^2 + cx + ab = 0$.
Statement-$2$: $a + b + c = 0$.