The equation 3x^2 – 5x + 4 = 0 has roots P and Q, find a quadratic equation with the roots (P + 1/2Q) and (Q + 1/2P)

We know the roots of the equation 3x² - 5x + 4 = 0 is P & Q, therefore is is equivalent to (x - P)(x - Q) = 0. Expanding the expression we get x² - x(P+Q) + PQ = 0. Equating coefficents with the original we see that P + Q = 5/3 & PQ = 4/3. The equation of a quadratic is (x- c₁)(x-c₂). Let c₁ = P + 1/2Q & c₂ = Q + 1/2P to get [x - (P + 1/2Q)][x - (Q + 1/2P)] = 0. Expanding and simplifying gives x² - x(P + 1/2Q + Q + 1/ 2P) + (P +1/2Q)(Q + 1/2P) = 0, Simplifiying further gives x² -x[(P+Q) +((P+Q)/(2PQ))] + (PQ +1 1/4PQ) = 0. Substituting P + Q = 5/3 & PQ = 4/3 to get x² - x(55/24) + 121/48 = 0. Multiply everything by 48 to get integer values and a final answer 48x² - 110x + 121 = 0. (did not include long calculation in paragraph as it is not very clear when typed)