This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
a) Step 1: Identify the sum and product of the roots of the given quadratic equation. The given quadratic equation is 2x^2 - 5x + k = 0. The roots are x_1 = 1+ and x_2 = 1+. For a quadratic equation ax^2+bx+c=0, the sum of roots is -(b)/(a) and the product of roots is (c)/(a). Sum of roots: (1+) + (1+) = -(-5)/(2) 2 + + = (5)/(2) + = (5)/(2) - 2 + = (1)/(2) Product of roots: (1+)(1+) = (k)/(2) 1 + + + = (k)/(2) Substitute + = (1)/(2): 1 + (1)/(2) + = (k)/(2) (3)/(2) + = (k)/(2) Multiply by 2: 3 + 2 = k 2 = k-3 = (k-3)/(2) Step 2: Find the sum and product of the roots of the new quadratic equation. The roots of the new quadratic equation are y_1 = 1+^2 and y_2 = 1+^2. Let S' be the sum of the new roots and P' be the product of the new roots. Sum of new roots S': S' = (1+^2) + (1+^2) S' = 2 + ^2 + ^2 We know that ^2 + ^2 = (+)^2 - 2. Substitute the values of + and : ^2 + ^2 = ((1)/(2))^2 - 2((k-3)/(2)) ^2 + ^2 = (1)/(4) - (k-3) ^2 + ^2 = (1)/(4) - k + 3 ^2 + ^2 = (1+12)/(4) - k ^2 + ^2 = (13)/(4) - k Now substitute this back into the expression for S': S' = 2 +