Solve by Factoring x^4-20x^2+64=0

Problem

x4−20*x2+64=0

Solution

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

u2−20*u+64=0

2. Factor the quadratic expression by finding two numbers that multiply to 64 and add to −20 These numbers are −16 and −4

(u−16)*(u−4)=0

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

(x2−16)*(x2−4)=0

4. Factor the resulting expressions using the difference of squares formula, a2−b2=(a−b)*(a+b)

(x−4)*(x+4)*(x−2)*(x+2)=0

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

x−4=0⇒x=4

x+4=0⇒x=−4

x−2=0⇒x=2

x+2=0⇒x=−2

Final Answer

x={−4,−2,2,4}

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