How do you simplify #x^2/(x^2-4) = x/(x+2)-2/(2-x) #?

Question

Isaiah Anthony · Accepted Answer

The given equation represents an impossible relation,
...unless (see below)

If we attempt to simplify the given equation (by multiplying both sides by #(x^2-4)# with the assumption #x !in {-2,+2}#

#x^2=x(x-2)+(2(x+2)#

#color(red)(cancel(color(black)(x^2)))=color(red)(cancel(color(black)(x^2)))color(blue)(cancel(color(black)(-2x)))color(blue)(cancel(color(black)(+2x)))+4#

#0=4color(white)("XXXX")#Impossible!

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As an alternative answer to the one given above, it is possible to claim a pair of solutions: #color(magenta)(x=-2)# or #color(magenta)(x=+2)#

and if we look at the graphs for the left and right sides of the given equation this (sort of) makes sense:

Genesis Allen · Answer

undefined To simplify the equation x^2/(x^2-4) = x/(x+2)-2/(2-x), we can start by finding a common denominator for the fractions on the right side of the equation. The common denominator is (x+2)(2-x).

Next, we can rewrite the fractions on the right side with the common denominator:

x/(x+2) = (x(2-x))/((x+2)(2-x))
-2/(2-x) = (-2(x+2))/((x+2)(2-x))

Now, we can substitute these values back into the original equation:

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

Simplifying further, we can cancel out the common factors:

x^2 = x(2-x) + 2(x+2)

Expanding and simplifying the equation:

x^2 = 2x - x^2 + 2x + 4

Combining like terms:

2x^2 - 4x - 4 = 0

Factoring the quadratic equation:

2(x-2)(x+1) = 0

Setting each factor equal to zero:

x-2 = 0 or x+1 = 0

Solving for x:

x = 2 or x = -1

Therefore, the simplified solution to the equation x^2/(x^2-4) = x/(x+2)-2/(2-x) is x = 2 or x = -1.