Solving A Cool Rational Equation

[@SidneiMV]

Brilliant! 👏🏼👏🏼👏🏼

[@jalalal-misrati9045]

Nice master, thanks

[@pwmiles56]

The equation can be written x^2+(-3x/(x+3))^2 = 16 Notice that -3x/(x+3) is involutive . This happens if in f(x)=(ax+b)/(cx+d), a+d is zero. Hence if x is a solution, so is -3x/(x+3) Therefore the roots of the quartic can be written x1, x2, -3x1/(x1+3), -3x2/(x2+3) Multiplying out and expanding the quartic is x^2(x+3)^2 + 9x^2 - 16(x+3)^2 = 0 x^2(x^2+6x+9) + 9x^2 - 16(x^2+6x+9) = 0 x^4 + 6x^3 + 2x^2 - 96x - 144=0 By Vieta formulas 9x1^2 x2^2 /((x1+3)(x2+3)) = -144 x1^2 x2^2 /((x1+3)(x2+3)) = -16 x1 + x2 - 3x1/(x1+3) - 3x2/(x2+3) = -6 Combining terms in x1 and x2 x1^2/(x1+3) + x2^2/(x2+3) = -6 Put u = x1^2/(x1+3) v = x2^2/(x2+3) We get uv = -16 u + v = -6 (u-v)^2 = (u+v)^2 - 4uv = 36+64 = 100 u - v = +/-10 u = 2, v=-8 (or vice versa) Solve x1^2/(x1+3) = 2 x1^2 - 2x1 - 6 = 0 x1 = (2+/-sqrt(28))/2 x1 = 1 +/- sqrt(7) x2^2/(x2+3) = -8 x2^2 + 8x2 + 24 = 0 x2 = (-8+/-sqrt(64-96))/2 x2 = -4 +/- 2 i sqrt(2)

[@Augustus9720]

let x+3=t,x=t-3so the equation turn to t⁴-6t³+2t-54t+81=0,Factor:(t²-8t+9)(t²+2t+9)=0,t=4±√7 or -1±2√2i, so x=1±√7 or -4±2√2i

[@MrGeorge1896]

Another way is to solve the quartic equation directly: x⁴ + 6x³ + 2x² - 96x - 144 = 0 (x² + ax + b) (x² - (a - 6)x - 144/b) = 0 This is somewhat cumbersome as 144 has so many factors but with b = 24 (and so a = 8) we can crack this.

[@FisicTrapella]

Actually, it is a parabola (x^2) modified by an extra term.

[@miguelgnievesl6882]

X=1±√7.

[@RyanLewis-Johnson-wq6xs]

X^2+(9X^2)/(X+3)^2=16 X=1±Sqrt[7] X=-4±2Sqrt[2]i Final Answer

[@phill3986]

Yeah think he missed the square root on the 7 the first time