Solve by Factoring x^4-7x^2+6=0

Problem

x4−7*x2+6=0

Solution

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

u2−7*u+6=0

2. Factor the quadratic expression by finding two numbers that multiply to 6 and add to −7 These numbers are −1 and −6

(u−1)*(u−6)=0

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

(x2−1)*(x2−6)=0

4. Factor further using the difference of squares for the first term, (x2−1)=(x−1)*(x+1)

(x−1)*(x+1)*(x2−6)=0

5. Set each factor to zero to solve for x

x−1=0⇒x=1

x+1=0⇒x=−1

x2−6=0⇒x2=6⇒x=±√(,6)

Final Answer

x=1,−1,√(,6),−√(,6)

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