Solve by Factoring x^4-3x^2+2=0

Problem

x4−3*x2+2=0

Solution

1. Identify the equation as a quadratic-type equation by letting u=x2

2. Substitute u into the equation to rewrite it in terms of u

u2−3*u+2=0

3. Factor the quadratic expression by finding two numbers that multiply to 2 and add to −3

(u−1)*(u−2)=0

4. Substitute back x2 for u to return to the original variable.

(x2−1)*(x2−2)=0

5. Factor further using the difference of squares for the first term and treating the second term as a difference of squares involving a square root.

(x−1)*(x+1)*(x−√(,2))*(x+√(,2))=0

6. Apply the zero product property by setting each factor equal to zero and solving for x

x−1=0⇒x=1

x+1=0⇒x=−1

x−√(,2)=0⇒x=√(,2)

x+√(,2)=0⇒x=−√(,2)

Final Answer

x={1,−1,√(,2),−√(,2)}

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