Maths about algebra and polynomials

Question

If X+Y=86 and X²-X⁴=82 then find the value of X-Y

Answer 1

We first solve the fourth degree equation in quadratic form for x^2. x^4 - x^2 +82 = 0. By the quadratic formula, x^2 = (1 +- sqrt (1 - 4 * 82))/2 = ( 1+- i sqrt (327))/2. To find the square root of the complex number, we convert to polar form, r = sqrt ( (1/2)^2 + (Sqrt(327)./2)^2) = sqrt ( 1/4 + 327/4) = sqrt (81) = 9. theta = arctan (sqrt (327)) =86.835 degrees. To find the square root, we take the square root of r and have of theta. So x = cis (3, 43.417) = 3 * cos (43.417) + i *3 * sin (43.417) = 2.18 + 2.06i. y= 86 - x = 86 - (2.18 + 2.06 i) = 83.82 -2.06i. x-y = 2.18 + 2.06i - (83.82 - 2.06i) = -81.64 + 4.12 i.

This was an interesting and reasonably difficult problem.