Solve by Factoring x^4-37x^2+36=0

Problem

x^4−37*x^2+36=0

Solution

1. Substitute a new variable to treat the equation as a quadratic. Let u=x^2 which implies u^2=x^4

u^2−37*u+36=0

2. Factor the quadratic expression by finding two numbers that multiply to 36 and add to −37 These numbers are −36 and −1

(u−36)*(u−1)=0

3. Back-substitute x^2 for u to return to the original variable.

(x^2−36)*(x^2−1)=0

4. Apply the difference of squares formula, a^2−b^2=(a−b)*(a+b) to both factors.

(x−6)*(x+6)*(x−1)*(x+1)=0

5. Solve for x by setting each linear factor equal to zero.

x−6=0⇒x=6

x+6=0⇒x=−6

x−1=0⇒x=1

x+1=0⇒x=−1

Final Answer

x={6,−6,1,−1}

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