Solving x^6+x^4+x^2=3 | Math Olympiads

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[shainypd]

Sybermath I am ur biggest fan. Thx for making such amazing puzzles 🙂🙂🙂🙂

[alextang]

It is obviously that we can sub u=x^2.
After sub, equation changed to 3rd degree
It is then obviously that u=1 is a root.
It now is down to 2nd degree, so we can use quadratic formula to solve it.

[VinTheFox]

You missed two of the four complex solutions and the two you gave can be simplified greatly! ±√[(√3-1)/2] ± i√[(√3+1)/2]

[Umbra]

Substitute u=x^2 to give us u^3+u^2+u = 3. By inspection, u=1 is a solution, and since the left-hand-side is increasing for all possible values of u (as u is a non-negative number) it is the only real solution. Factoring (u-1) from u^3+u^2+u-3 (a rewritten version of the above equation) gives us u^2 + 2u + 3, which factors to give us u = -1 +- sqrt(2)i. Getting back to x from there requires more math than I’m willing to do on a Tuesday morning. x = +- 1 are two solutions.

[thomaswaynejunior]

having figured out –1 and 1 are of the solutions, factorize the polynomial using horner's method

[spikepls]

X = ±1
i solved it instantly when seen

[timehorse]

You forgot 2 solutions, the cases where u uses the negative square root!

[scottleung]

My complex solutions came out as +-sqrt(i*sqrt(2)-1) and +-i*sqrt(i*sqrt(2)+1) without using trig. If this question came up on a test and you saw these answers, would they be acceptable?

[rakenzarnsworld]

X^3+X^2+X=3
X=1
x^2=X, x=+-1
(X-1)(X^2+2X+3)=0
X=-1+-(1.414)i

[HoSza]

The form of the complex roots are horribly over complicated, if you insist on choosing the trigonometric method to calculate the square roots, at least simplify the result: ±√((√3-1)/2)±i√((√3+1)/2). See? No ugly 4th roots and square roots in the denominators. The algebraic method gives this result directly by the way.

[kmsbean]

obviously x = 1, -1, then even if u = x^2 gets complex and I don't feel like typing out the other 2 u let alone the more complex 4 x sqrt of i sqrt

[mihaleben]

Its, really easy, its 1
Yeah, illusion of difficulty