Solve by Factoring x^4-16=0

Problem

x^4−16=0

Solution

1. Recognize the difference of squares on the left side of the equation, noting that x^4=(x^2)^2 and 16=4^2

x^4−16=(x^2−4)*(x^2+4)

2. Factor further by recognizing that x^2−4 is also a difference of squares, where 4=2^2

(x−2)*(x+2)*(x^2+4)=0

3. Apply the zero product property by setting each individual factor equal to zero.

x−2=0

x+2=0

x^2+4=0

4. Solve for x in each resulting equation to find all real and complex roots.

x=2

x=−2

x^2=−4⇒x=±2*i

Final Answer

x=2,−2,2*i,−2*i

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