How do you solve #x^4-18x^2+81=0#?
Refer to explanation
It is easy to see that
Hence we have that
Be aware that roots
because we have a fourth degree polynomial.
Normally, to solve a polynomial of degree 4 like the one here, you need to do synthetic division and use a lot of theorems and rules - it gets kinda messy. However, this one is special because we can actually make it a quadratic equation.
We do this by letting
Doesn't that look better? Now we're dealing with a nice, easy quadratic equation. In fact, this is a perfect square; in other words, when you factor it, you get
To solve, we take the square root of both sides:
And this simplifies to
Finally, we add 9 to both sides to get
Awesome! Almost there. However, our original problem has
Therefore, our solutions are