Add (x-4)/(x^2-2x)+4/(x^2-4)

Math
x-4×2-2x+4×2-4
Simplify each term.
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Factor x out of x2-2x.
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Factor x out of x2.
x-4x⋅x-2x+4×2-4
Factor x out of -2x.
x-4x⋅x+x⋅-2+4×2-4
Factor x out of x⋅x+x⋅-2.
x-4x(x-2)+4×2-4
x-4x(x-2)+4×2-4
Simplify the denominator.
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Rewrite 4 as 22.
x-4x(x-2)+4×2-22
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=x and b=2.
x-4x(x-2)+4(x+2)(x-2)
x-4x(x-2)+4(x+2)(x-2)
x-4x(x-2)+4(x+2)(x-2)
To write x-4x(x-2) as a fraction with a common denominator, multiply by x+2x+2.
x-4x(x-2)⋅x+2x+2+4(x+2)(x-2)
To write 4(x+2)(x-2) as a fraction with a common denominator, multiply by xx.
x-4x(x-2)⋅x+2x+2+4(x+2)(x-2)⋅xx
Write each expression with a common denominator of x(x-2)(x+2), by multiplying each by an appropriate factor of 1.
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Multiply x-4x(x-2) and x+2x+2.
(x-4)(x+2)x(x-2)(x+2)+4(x+2)(x-2)⋅xx
Multiply 4(x+2)(x-2) and xx.
(x-4)(x+2)x(x-2)(x+2)+4x(x+2)(x-2)x
Reorder the factors of x(x-2)(x+2).
(x-4)(x+2)x(x+2)(x-2)+4x(x+2)(x-2)x
Reorder the factors of (x+2)(x-2)x.
(x-4)(x+2)x(x+2)(x-2)+4xx(x+2)(x-2)
(x-4)(x+2)x(x+2)(x-2)+4xx(x+2)(x-2)
Combine the numerators over the common denominator.
(x-4)(x+2)+4xx(x+2)(x-2)
Simplify the numerator.
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Expand (x-4)(x+2) using the FOIL Method.
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Apply the distributive property.
x(x+2)-4(x+2)+4xx(x+2)(x-2)
Apply the distributive property.
x⋅x+x⋅2-4(x+2)+4xx(x+2)(x-2)
Apply the distributive property.
x⋅x+x⋅2-4x-4⋅2+4xx(x+2)(x-2)
x⋅x+x⋅2-4x-4⋅2+4xx(x+2)(x-2)
Simplify and combine like terms.
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Simplify each term.
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Multiply x by x.
x2+x⋅2-4x-4⋅2+4xx(x+2)(x-2)
Move 2 to the left of x.
x2+2⋅x-4x-4⋅2+4xx(x+2)(x-2)
Multiply -4 by 2.
x2+2x-4x-8+4xx(x+2)(x-2)
x2+2x-4x-8+4xx(x+2)(x-2)
Subtract 4x from 2x.
x2-2x-8+4xx(x+2)(x-2)
x2-2x-8+4xx(x+2)(x-2)
Add -2x and 4x.
x2+2x-8x(x+2)(x-2)
Factor x2+2x-8 using the AC method.
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Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -8 and whose sum is 2.
-2,4
Write the factored form using these integers.
(x-2)(x+4)x(x+2)(x-2)
(x-2)(x+4)x(x+2)(x-2)
(x-2)(x+4)x(x+2)(x-2)
Cancel the common factor of x-2.
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Cancel the common factor.
(x-2)(x+4)x(x+2)(x-2)
Rewrite the expression.
x+4x(x+2)
x+4x(x+2)
Add (x-4)/(x^2-2x)+4/(x^2-4)

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