Solve by Factoring x^4-6x^2+8=0

Problem

x4−6*x2+8=0

Solution

1. Substitute a new variable to treat the equation as a quadratic. Let u=x2 which implies u2=x4

u2−6*u+8=0

2. Factor the quadratic expression by finding two numbers that multiply to 8 and add to −6 These numbers are −2 and −4

(u−2)*(u−4)=0

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

(x2−2)*(x2−4)=0

4. Factor further using the difference of squares for the second term, where x2−4=(x−2)*(x+2)

(x2−2)*(x−2)*(x+2)=0

5. Set each factor to zero to solve for x

x2−2=0⇒x2=2⇒x=±√(,2)

x−2=0⇒x=2

x+2=0⇒x=−2

Final Answer

x=±√(,2),±2

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