I"m supposed to solve this equation. It"s from a hyakkendana-hashigozake.com contest so solving it by hand would be preferable (no quartic formulas).I thought about making \$u = x^2-3x-2\$ obviously but it leads to another quartic equation. I also tried the substitution \$u=x+2\$, and after the whole expand trinomial, simplify, invoke rational root theorem and test roots, I still got nothing out of it.

You are watching: (3x^2)^2

I noticed that \$x^2-3x-2\$ can"t be factored nicely so I dunno what other route to take. Lots of equations I tackled in hyakkendana-hashigozake.com contests could make use of nice trigonometric substitutions, but none in particular pop in my head right now.

If anyone can give me hints or a full solution, that would be awesome. Thanks!

Since you have a quartic, there are potentially 4 real solutions.

Note that any solution to \$x = x^2 - 3x - 2\$ is also a solution to the given equation.

Hence, \$x^2 - 4x - 2\$ is a factor of the quartic. Now find the other factor, and solve both quadratics.

Alternatively, see this almost 10 year old thread on the Art of Problem Solving Forum, or this slightly newer thread which discusses a similar problem.

Thanks for contributing an answer to hyakkendana-hashigozake.comematics Stack Exchange!

Please be sure to answer the question. Provide details and share your research!

But avoid

Asking for help, clarification, or responding to other answers.Making statements based on opinion; back them up with references or personal experience.

Use hyakkendana-hashigozake.comJax to format equations. hyakkendana-hashigozake.comJax reference.

See more: 17 Battery Place New York Ny 10004, 17 Battery South

To learn more, see our tips on writing great answers.

## Not the answer you're looking for? Browse other questions tagged algebra-precalculus polynomials contest-hyakkendana-hashigozake.com or ask your own question.

Solve \$sqrt1 + sqrt1-x^2left(sqrt(1+x)^3 + sqrt(1-x)^3 ight) = 2 + sqrt1-x^2 \$

site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. rev2021.11.12.40742

hyakkendana-hashigozake.comematics Stack Exchange works best with JavaScript enabled