# How do you solve #x^4-18x^2+81=0#?

##### 2 Answers

Refer to explanation

#### Explanation:

It is easy to see that

Hence we have that

Be aware that roots

because we have a fourth degree polynomial.

#### Explanation:

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