Solve by Factoring x^4=81

Problem

x^4=81

Solution

1. Rearrange the equation to set it equal to zero by subtracting 81 from both sides.

x^4−81=0

2. Identify the expression as a difference of squares, since x^4=(x^2)^2 and 81=9^2

(x^2)^2−9^2=0

3. Factor using the difference of squares formula a^2−b^2=(a−b)*(a+b)

(x^2−9)*(x^2+9)=0

4. Factor again the first term (x^2−9) which is also a difference of squares.

(x−3)*(x+3)*(x^2+9)=0

5. Set each factor to zero to find the real and imaginary solutions.

x−3=0⇒x=3

x+3=0⇒x=−3

x^2+9=0⇒x^2=−9⇒x=±3*i

Final Answer

x=3,−3,3*i,−3*i

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