If x, y and z are real and different and $$\,u = {x^2} + 4{y^2} + 9{z^2} - 6yz - 3zx - 2xy$$, then u is always.

The equation x + 2y + 2z = 1 and 2x + 4y + 4z = 9 have

If $$\ell $$, m, n are real, $$\ell \ne m$$, then the roots by the equation :
$$(\ell - m)\,{x^2} - 5\,(\ell + m)\,x - 2\,(\ell - m) = 0$$ are

If $$\alpha ,\,\beta $$ are the roots of $${x^2} + px + q = 0$$ and $$\gamma ,\,\delta $$ are the roots of $${x^2} + rx + s = 0,$$ evaluate $$\left( {\alpha - \gamma } \right)\left( {\alpha - \delta } \right)\left( {\beta - \gamma } \right)$$ $$\left( {\beta - \delta } \right)$$ in terms of $$p,\,q,\,r$$ and $$s$$.

deduce the condition that the equations have a common root.