if 2x-1/2x=13 then find the value of 16x^4+1/16x^4
Answers
Multiplying both sides of the equation by x, we get 2x^2 -1 = 6x, then subtracting 6x from both sides: 2x^2 - 6x -1 = 0. Applying the quadratic formula with a = 2, b = -6 and c = -1, x = [6 ± √(6^2 - 4*2*(-1))]/(2*2) = [6 ± √(36 + 8)]/4 = [6 ± √44]/4 = [6 ± √(4*11)]/4 = [6 ± 2√11]/4 = 3/2 ± √11/2. Let's use the positive value of x (since 3^2 = 9 < 11, the negative square root gives a negative value for x). x^2 = (3/2 + √11/2)^2 = 9/4 + 3√11/2 + 11/4 = 20/4 + 3√11/2 = 5 + 3√11/2, so x^4 = (5 + 3√11/2)^2 = 25 + 2*5*3√11/2 + 9*11/4 = 100/4 + 99/4 + 15√11 = 199/4 + 15√11, so 16x^4 + 1/x^4 = 16(199/4 + 15√11) + 1/(199/4 + 15√11) = 4*199 + 16*15√11 + (199/4 - 15√11)/(199/4 + 15√11)(199/4 - 15√11) = 796 + 240√11 + (199/4 - 15√11)/[(199/4)^2 - (15^2)*11] = 796 + 240√11 + (199/4 - 15√11)/[39601/16 - 225*11] = 796 + 240√11 + (199/4 - 15√11)/[39601/16 - 2475] = 796 + 240√11 + (199/4 - 15√11)/[(39601 - 39600)/16] = 796 + 240√11 + 16(199/4 - 15√11) = 796 + 240√11 + 4*199 - 16*15√11 = 796 + 240√11 + 796 - 240√11 = 1592