Question

question 4 · 1 point
find all the solutions, including imaginary solutions if they exist, of the following equation.
$x^{4}+20x^{2}+64 = 0$
list your answers separated by commas, not using a ± sign.
for example, if you found that $x=pm1$ or $x = pm i$, you would enter 1, - 1, i, - i.
provide your answer below:

Explanation:

Step1: Let \(u = x^{2}\)

Substitute \(u\) into the equation: \(u^{2}+20u + 64=0\)

Step2: Factor the quadratic equation

\((u + 4)(u + 16)=0\)

Step3: Solve for \(u\)

Using the zero - product property: \(u+4 = 0\) gives \(u=-4\); \(u + 16=0\) gives \(u=-16\)

Step4: Substitute back \(u = x^{2}\)

For \(u=-4\), we have \(x^{2}=-4\), so \(x=\pm2i\); for \(u=-16\), we have \(x^{2}=-16\), so \(x=\pm4i\)

Answer:

\(2i,-2i,4i,-4i\)