Question

solve.
$x^{4}+12x^{2}+20 = 0$
write your answer as a list of values separated by commas. simplify all radicals.

Explanation:

Step1: Let \( y = x^2 \), then the equation becomes \( y^2 + 12y + 20 = 0 \)

We substitute \( x^2 \) with \( y \) to convert the quartic equation into a quadratic equation, which is easier to solve.

Step2: Solve the quadratic equation \( y^2 + 12y + 20 = 0 \) by factoring

We need to find two numbers that multiply to 20 and add up to 12. The numbers are 10 and 2. So, we can factor the equation as \((y + 10)(y + 2) = 0\)

Step3: Set each factor equal to zero and solve for \( y \)

For \( y + 10 = 0 \), we get \( y = - 10 \). For \( y + 2 = 0 \), we get \( y = - 2 \)

Step4: Substitute back \( y = x^2 \) and solve for \( x \)

• When \( y = - 10 \), we have \( x^2 = - 10 \). Using the definition of the imaginary unit \( i = \sqrt{-1} \), we get \( x = \pm\sqrt{-10}=\pm i\sqrt{10} \)
• When \( y = - 2 \), we have \( x^2 = - 2 \). Using the definition of the imaginary unit \( i = \sqrt{-1} \), we get \( x = \pm\sqrt{-2}=\pm i\sqrt{2} \)

Answer:

\(i\sqrt{10}, -i\sqrt{10}, i\sqrt{2}, -i\sqrt{2}\)