Q5 - Equations with an "additonal" restricted variable 1. The problem statement, all variables and given/known data P1 Determine the parameter m such that the sum of the squares of the roots of equation x2 - (m-2)x + m - 3 = 0 is minimal. P2 Determine the values of parameter p, so that the solution of equation p - (1/x) = (p2 - 2x - 5)/((p+3)x) is greather than 1. 2. Relevant equations 3. The attempt at a solution I would like to know whether there is any specific way of calculating equations with an additonal "restricted" variable, like in the two problems above. Is a rational reasoning followed by a trial and error used or is more specific calculation possible? P1 x2 - (m-2)x + m - 3 = 0 The sum of the squares must be minimal, thus: (ax - b)(cx - d) where ax2 + cx2 must be minimal roots are minimal when (x - 1)2 or (x + 1)2 namely 12 or (-1)2 which equals 1 This form x2 - 2x +1 x2 +2x +1 Can be achieved when x2 - (4-2)x + 4 - 3 = 0 Thus m = 4 Are my calculations here correct? Is this the correct way to solve this problem? P2 p - (1/x) = (p2 - 2x - 5)/((p+3)x) I find this problem a lot more difficult. Reasoning here is a lot trickier (for me at least). I can only see that p can not equal -3 When either working the equation out of fractions, ie (px - 1)(p +3) = p2 - 2x - 5 or moving everything to one side, I am still left with one equation and "two" unknowns. Another approach I tried when anaylizing this problem was simply putting in a value for x, x=1, x=2 and x=3 all return values of p=-2 How should I approach such a problem?