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
Step 1: Simplify the expressions for x and y. We are given x = sqrt(2)-sqrt(3)sqrt(2)+sqrt(3) and y = sqrt(2)+sqrt(3)sqrt(2)-sqrt(3). Notice that y = (1)/(x). First, rationalize the denominator for x: x = sqrt(2)-sqrt(3)sqrt(2)+sqrt(3) × sqrt(2)-sqrt(3)sqrt(2)-sqrt(3) x = (sqrt(2)-sqrt(3))^2(sqrt(2))^2 - (sqrt(3))^2 x = 2 - 2sqrt(2)sqrt(3) + 32 - 3 x = 5 - 2sqrt(6)-1 x = -5 + 2sqrt(6) Since y = (1)/(x), we can also find y: y = (1)/(-5 + 2sqrt(6)) × -5 - 2sqrt(6)-5 - 2sqrt(6) y = -5 - 2sqrt(6)(-5)^2 - (2sqrt(6))^2 y = -5 - 2sqrt(6)25 - (4 × 6) y = -5 - 2sqrt(6)25 - 24 y = -5 - 2sqrt(6) Step 2: Calculate xy. Since y = (1)/(x), their product is: xy = x · (1)/(x) = 1 Step 3: Calculate x^2 and y^2. x^2 = (-5 + 2sqrt(6))^2 x^2 = (-5)^2 + 2(-5)(2sqrt(6)) + (2sqrt(6))^2 x^2 = 25 - 20sqrt(6) + 4 × 6 x^2 = 25 - 20sqrt(6) + 24 x^2 = 49 - 20sqrt(6) y^2 = (-5 - 2sqrt(6))^2 y^2 = (-5)^2 + 2(-5)(-2sqrt(6)) + (-2sqrt(6))^2 y^2 = 25 + 20sqrt(6) + 4 × 6 y^2 = 25 + 20sqrt(6) + 24 y^2 = 49 + 20sqrt(6) Step 4: Substitute the values into the expression 3x^2 - 5xy + 3y^2. 3x^2 - 5xy + 3y^2 = 3(49 - 20sqrt(6)) - 5(1) + 3(49 + 20sqrt(6)) = 147 - 60sqrt(6) - 5 + 147 + 60sqrt(6) = (147 + 147 - 5) + (-60sqrt(6) + 60sqrt(6)) = 294 - 5 + 0 = 289 The value of 3x^2 - 5xy + 3y^2 is 289. Send me the next one 📸