Given that \(x_1\) and \(x_2\) are the roots of \(ax^2 + bx + c = 0\), state in terms of some or all of \(a\), \(b\), \(c\): (i) the condition that \(x_1 = x_2\), (ii) the value of \(x_1 + x_2\).
Find the values of \(m\) for which the line \(y = mx\) is a tangent to the curve \(y^2 = 3x - 1\).
- The line \(y = 2x\) meets the curve \(3y = x^2 - 10\) at the points \(A(x_1, y_1)\) and \(B(x_2, y_2)\).
- Obtain the quadratic equation whose roots are \(x_1\) and \(x_2\).
- Without solving this equation, find the \(x\) co-ordinate of the midpoint of \(AB\).