Question

Find all the zeros of the equation
-3x^4 + 27x^2 + 1200 = 0

Answer 1

Answer:

$\displaystyle x=-5,\ x=5,\ x=4i,\ x=-4i$

Step-by-step explanation:

Biquadratic Equation

It's a fourth-degree equation where the terms of degree 1 and 3 are missing. It can be solved for the variable squared as if it was a second-degree equation, and then take the square root of the results

Our equation is

$\displaystyle -3x^4+27x^2+1200=0$

If we call $y=x^2$, our equation becomes a second-degree equation

$\displaystyle -3y^2+27y+1200=0$

Dividing by -3

$\displaystyle y^2-9y-400=0$

Factoring

$\displaystyle (y-25)(y+16)=0$

It leads to these solutions

$\displaystyle y=25\ ,\ y=-16$

Taking back the change of variable, we have for the first solution

$\displaystyle x^2=25\Rightarrow x=-5,x=5$

Now for the second solution, we get imaginary (complex) values

$\displaystyle x^2=-16\Rightarrow x=4i,\ x=-4i$

Summarizing, the four solutions for x are

$\displaystyle x=-5,\ x=5,\ x=4i,\ x=-4i$