Solve for x

Problem

1/(x4+4)+1/(x4−4*x3+4*x2−4)−2/(x4−4*x2−8*x−4)=0

Solution

1. Factor the first denominator using Sophie Germain's Identity by adding and subtracting 4*x2

x4+4=(x2+2)2−(2*x)2=(x2+2*x+2)*(x2−2*x+2)

2. Factor the second denominator by grouping terms to reveal a difference of squares.

x4−4*x3+4*x2−4=(x2−2*x)2−2=(x2−2*x+2)*(x2−2*x−2)

3. Factor the third denominator by grouping terms to reveal a difference of squares.

x4−(4*x2+8*x+4)=(x2)2−(2*x+2)2=(x2+2*x+2)*(x2−2*x−2)

4. Rewrite the equation using the factored forms and find a common denominator, which is (x2+2*x+2)*(x2−2*x+2)*(x2−2*x−2)

1/((x2+2*x+2)*(x2−2*x+2))+1/((x2−2*x+2)*(x2−2*x−2))−2/((x2+2*x+2)*(x2−2*x−2))=0

5. Combine the fractions over the common denominator.

((x2−2*x−2)+(x2+2*x+2)−2*(x2−2*x+2))/((x2+2*x+2)*(x2−2*x+2)*(x2−2*x−2))=0

6. Simplify the numerator by distributing and combining like terms.

x2−2*x−2+x2+2*x+2−2*x2+4*x−4=0

4*x−4=0

7. Solve for x and check for extraneous solutions by ensuring the denominators are non-zero.

4*x=4

x=1

Final Answer

1/(x4+4)+1/(x4−4*x3+4*x2−4)−2/(x4−4*x2−8*x−4)=0⇒x=1

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