Solve the equaltion for x.

Question

5(5x²−17)²−343x−833=0

Answer 1

5(5x^2 - 17)^2 - 343x -833 = 0

5(25x^4 - 170x^2 + 289) -343x - 833 = 0

125x^4 -850x^2 + 1445 -343x - 833 = 0

125x^4 - 850x^2 -343x + 612 = 0

It is degree 4 and has 4 solutions

by Descartes' Rule of signs there are max 2 positive real solutions, and max 2 negative real solutions

x f(x)

0 612

1 -450 that change in sign means there's one solution between x=0 and x= 1 one solution is a positive proper fraction, 0<x<1

eventually the highest degree dominates and there's another solution with x>>1

that's the two real positive solutions

Now, it's either 2 negative real solutions or 2 imaginary solutions. They come in conjugate pairs for the imaginary ones.

0 612

-1 230

-2 538

-3 >>538

it doesn't look good for a negative real solution. Probably 2 imaginary solutions

Solving cubic equations is hard enough. 4th degree can be worse.