Break down $x^4 + 5x^2 +5$

Question

How do I break down the function in the title even further? I think that I need to use a square root somewhere, but I'm not certain.

Answer 1

you can make a change of variable $x^2 = u$ to turn $x^4 + 5x^2 + 5 = 0$ into the quadratic equation $$u^2 + 5u + 5 = 0. $$ this has the solution $$u = \frac{-5 \pm \sqrt{25 - 4 \times 5}}{2} = \frac{-5 \pm \sqrt 5}{2}$$ here we see that both roots are negative. therefore $x^2 = u$ has no real solutions consequently $$x^4 + 5x^2 +5 = 0 $$ has no real solution for $x.$

p.s. if you wanted to factor the expression, then we can use $$u^2 + 5u + 5 = \frac 14(2u + 5 + \sqrt 5)(2u + 5 - \sqrt5) \to\\ x^4 + 5x^2 + 5 = \frac 14(2x^2 + 5 + \sqrt 5)(2x^2 + 5 - \sqrt5) $$