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
Right mary, let's go. Here are the evaluations for question 2a: Given the quadratic equation 2x^2 - 4x + 7 = 0. For a quadratic equation ax^2 + bx + c = 0, the sum of the roots is + = -(b)/(a) and the product of the roots is = (c)/(a). In this case, a=2, b=-4, c=7. Sum of roots: + = -(-4)/(2) = (4)/(2) = 2 Product of roots: = (7)/(2) i) Evaluate ()/() + ()/(): Step 1: Combine the fractions. ()/() + ()/() = (^2 + ^2)/() Step 2: Express ^2 + ^2 in terms of + and . ^2 + ^2 = ( + )^2 - 2 Step 3: Substitute the expressions and values. (( + )^2 - 2)/() = ((2)^2 - 2(7)/(2))(7)/(2) Step 4: Simplify the expression. (4 - 7)/(7)2 = (-3)/(7)2 = -3 × (2)/(7) = -(6)/(7) ii) Evaluate (1)/() + (1)/(): Step 1: Combine the fractions. (1)/() + (1)/() = ( + )/() Step 2: Substitute the values. (2)/(7)2 = 2 × (2)/(7) = (4)/(7) iii) Evaluate ^2 + ^2: Step 1: Use the identity ^2 + ^2 = ( + )^2 - 2. Step 2: Substitute the values. (2)^2 - 2((7)/(2)) = 4 - 7 = -3 iv) Evaluate ^3 + ^3: Step 1: Use the identity ^3 + ^3 = ( + )(^2 - + ^2). Step 2: Substitute ^2 + ^2 = ( + )^2 - 2. ^3 + ^3 = ( + )(( + )^2 - 2 - ) = ( + )(( + )^2 - 3) Step 3: Substitute the values. (2)((2)^2 - 3((7)/(2))) = 2(4 - (21)/(2)) Step 4: Simplify the expression. 2((8)/(2) - (21)/(2)) = 2(-(13)/(2)) = -13 v) Evaluate (1)/(^2) + (1)/(^2): Step 1: Combine the fractions. (1)/(^2) + (1)/(^2) = (^2 + ^2)/(()^2) Step 2: Substitute the value of ^2 + ^2 = -3 (from part iii) and = (7)/(2). (-3)/((7)2)^2 = (-3)/(49)4 Step 3: Simplify the expression. -3 × (4)/(49) = -(12)/(49) Send me the next one 📸