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
Given x = sqrt(2) + sqrt(3) and y = sqrt(2) - sqrt(3). a) Show that x^2 + y^2 = 4 Step 1: Calculate x^2. x^2 = (sqrt(2) + sqrt(3))^2 x^2 = (sqrt(2))^2 + 2(sqrt(2))(sqrt(3)) + (sqrt(3))^2 x^2 = 2 + 2sqrt(6) + 3 x^2 = 5 + 2sqrt(6) Step 2: Calculate y^2. y^2 = (sqrt(2) - sqrt(3))^2 y^2 = (sqrt(2))^2 - 2(sqrt(2))(sqrt(3)) + (sqrt(3))^2 y^2 = 2 - 2sqrt(6) + 3 y^2 = 5 - 2sqrt(6) Step 3: Calculate x^2 + y^2. x^2 + y^2 = (5 + 2sqrt(6)) + (5 - 2sqrt(6)) x^2 + y^2 = 5 + 2sqrt(6) + 5 - 2sqrt(6) x^2 + y^2 = 10 The statement x^2 + y^2 = 4 is incorrect for the given values of x and y. Instead, x^2 + y^2 = 10. b) Show that xy = 1 Step 1: Calculate xy. xy = (sqrt(2) + sqrt(3))(sqrt(2) - sqrt(3)) This is a difference of squares, (a+b)(a-b) = a^2 - b^2. xy = (sqrt(2))^2 - (sqrt(3))^2 xy = 2 - 3 xy = -1 The statement xy = 1 is incorrect for the given values of x and y. Instead, xy = -1. That's 2 down. 3 left today — send the next one.