Solve by Factoring x^4-81=0

Problem

x^4−81=0

Solution

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

2. Apply the formula for the difference of squares, a^2−b^2=(a−b)*(a+b) to the left side of the equation.

(x^2−9)*(x^2+9)=0

3. Factor further by recognizing that x^2−9 is also a difference of squares, where 9=3^2

(x−3)*(x+3)*(x^2+9)=0

4. Set each factor to zero using the zero product property to find the possible values for x

x−3=0⇒x=3

x+3=0⇒x=−3

x^2+9=0⇒x^2=−9

5. Solve for complex roots by taking the square root of both sides for the quadratic factor.

x=±√(,−9)

x=±3*i

Final Answer

x=3,−3,3*i,−3*i

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