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

Let $a, b \in R, a \neq 0$ be such that the equation $a x^{2}-2 b x+5=0$ has a repeated root $\alpha,$ which is also a root of the equation $x^{2}-2 b x-10=0$. If $\beta$ is the other root of this equation,then $\alpha^{2}+\beta^{2}$ is equal to

If $x = \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots \infty}}}$,then

Let $a, b, c$ be real numbers with $a \ne 0$. If $\alpha$ is a root of $a^2x^2 + bx + c = 0$,$\beta$ is a root of $a^2x^2 - bx - c = 0$,and $0 < \alpha < \beta$,then the equation $a^2x^2 + 2bx + 2c = 0$ has a root $\gamma$ that always satisfies:

The value of $k$ for which the equation $(k - 2)x^2 + 8x + (k + 4) = 0$ has both roots real,distinct,and negative is

If $l, m, n$ are real and $l \ne m$,then the roots of the equation $(l - m)x^2 - 5(l + m)x - 2(l - m) = 0$ are