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

Problem

x4−4*x2+3=0

Solution

1. Substitute a new variable to treat the equation as a quadratic. Let u=x2

u2−4*u+3=0

2. Factor the quadratic expression by finding two numbers that multiply to 3 and add to −4 These numbers are −1 and −3

(u−1)*(u−3)=0

3. Back-substitute x2 for u to return to the original variable.

(x2−1)*(x2−3)=0

4. Factor further using the difference of squares formula a2−b2=(a−b)*(a+b) for the first term, and treat 3 as (√(,3))2 for the second term.

(x−1)*(x+1)*(x−√(,3))*(x+√(,3))=0

5. Apply the Zero Product Property by setting each individual factor equal to zero and solving for x

x−1=0⇒x=1

x+1=0⇒x=−1

x−√(,3)=0⇒x=√(,3)

x+√(,3)=0⇒x=−√(,3)

Final Answer

x=1,−1,√(,3),−√(,3)

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