given that a and b are the roots a quadratic equation 2x2-2x+4 find a quadratic equation whose roots are a/b and b/a
+128473 2x^2 -2x + 4 = 0 divide through by 2
x^2 - x + 2 = 0 subtract 2 from both sides
x^2 - x = - 2 complete the square on x
x^2 - x + 1/4 = -2 + 1/4
(x - 1/2)^2 = -7/4 take both roots
x - 1/2 = ±i√7 / 2 add 1/2 to both sides
So
x = [ i√7 + 1] / 2 = a
a^2 = [ 1 + √7 i] [ 1 + √7 i ] / 4 = [ 1 + 2√7 i - 7] / 4 = [ √7 i - 3 ] / 2
Or
x = [ -i√7 + 1] / 2 = b
b^2 = [ 1 - √7 i ] [ 1 - √7 i ] / 4 = [ 1 -2√7 i - 7 ] / 4 = [ -√7 i - 3] / 2
a^2 + b^2 = -6/ 2 = -3
ab = [ 1 + i√7 ] / 2 * [ 1 - i√7 ] / 2 = [ 1 + 7] / 4 = 2
So ( x - a/b) (x - b/a) =
x^2 - (a/b)x - (b/a)x + (a/b)(b/a) =
x^2 - ( a/b + b/a)x + 1 =
x^2 - ( a^2 + b^2) x / ab + 1 =
x^2 - (-3)x / 2 + 1 =
x^2 + (3/2)x + 1
