(a) Roots of the equation = \(\frac{1}{2}\) and -7 \(\therefore (x - \frac{1}{2})(x + 7) = 0\) \(x^{2} - \frac{1}{2} x + 7x - \frac{7}{2} = 0\) \(x^{2} + \frac{13}{2} x - \frac{7}{2} = 0\) Multiply through by 2 to get integers as coefficients, \(2x^{2} + 13x - 7 = 0\) (b) Let the father and daughter's present ages be r and t respectively. Three years ago, father = (r - 3) years \((r - 3) = 4t \implies r = (4t + 3) ... (1)\) \(r \times t = 430 ... (2)\) Substitute (1) in (2), \((4t + 3) \times t = 430 \implies 4t^{2} + 3t = 430\) \(\implies 4t^{2} + 3t - 430 = 0\) \(4t^{2} - 40t + 43t - 430 = 0\) \(4t(t - 10) + 43(t - 10) = 0\) \(\therefore (4t + 43)(t - 10) = 0\) \(t = \frac{-43}{4}; t = 10\) Since t is age, it cannot be negative. Hence, t = 10 years. Recall : \(r = 4t + 3\) \(\therefore r = 4(10) + 3 = 40 + 3 = 43\) Therefore, the father's present age is 43 years while the daughter's age is 10 years.