Question

What are the real and complex solutions of the polynomial equation?

x^4 - 41x^2 = -400

Answer 1

Have you considered the Quadratic Formula?

Answer 2

let x^2 = a
solve quatratic then resubstitute

Answer 3

a^2 - b^2 = c^2
i = sqrt of -1

Answer 4

Example: \(\color{black}{\displaystyle x^4+2x^2-3=0 }\)
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We are going to substitute \(\color{black}{\displaystyle a=x^2 }\).

We get,
\(\color{black}{\displaystyle x^4+2x^2-3=0 }\)
\(\color{black}{\displaystyle (x^2)^2+2(x^2)-3=0 }\)
\(\color{black}{\displaystyle a^2+2a-3=0 }\)
\(\color{black}{\displaystyle (a-1)(a+3)=0 }\)
\(\color{black}{\displaystyle a=1,~-3 }\).

Then, since we had \(\color{black}{\displaystyle a=x^2 }\), we are to consider two cases:

\(\color{black}{\displaystyle x^2=1 }\) \(\color{black}{\displaystyle x^2=-3 }\)
\(\color{black}{\displaystyle x=\pm 1 }\) \(\color{black}{\displaystyle x=\pm \sqrt{3}i }\)

Answer 5

So for that example you would get \(\color{black}{\displaystyle x =-1,~~1,~~\sqrt{3}i,~~-\sqrt{3}i}\) .

Answer 6

This is a spicy one! Solution attached, use at your own discretion. The final answer implies four roots, which is consistent with our expectations considering the degree of the polynomial was four.

I have not simplified the final expression.

Answer 7

Simplifying the solution you'll get the nice and neat x values ±4, ±5