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

Question

Genesis Allen · Accepted Answer

#x=2#

Write as:

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

Using the principle that #color(white)("s") a^2-b^2=(a+b)(a-b)#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(brown)("The sort of cheating bit - not really!")# Consider #-2/(2-x)#.#color(white)(c)# We need to change this such that the #color(white)()# denominator is #x-2# if we whish to use the context of #color(white)()# #color(white)("s") a^2-b^2=(a+b)(a-b)#

Suppose we had #+2/(x-2)# which is the format we wish to have. If we use this is it another form of what was originally written. That is; does it have the same value?

#color(green)([2/(x-2)color(red)(xx1)] ->[2/(x-2)color(red)(xx(-1)/(-1))] = -2/((2-x))#

So #+2/(x-2)# has the same value as #-2/(2-x)# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Write as:

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

This is equivalent to:

#x^2/(x^2-2^2)=(x xx2)/((x+2)(x-2))#

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

We only need to think about the numerators because the denominators are the same.

#x^2=2x#

By x, divide both sides.

#x=2#

Abigail Allen · Answer

undefined To solve the equation x^2/(x^2-4) = x/(x+2) - 2/(2-x), we can follow these steps:

1. Start by finding a common denominator for the fractions on the right side of the equation. The common denominator is (x+2)(2-x).

2. 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)]

3. Simplify the fractions on the right side:
   x/(x+2) = (2x-x^2)/[(x+2)(2-x)]
   -2/(2-x) = -2(x+2)/[(x+2)(2-x)]

4. Combine the fractions on the right side:
   (2x-x^2)/[(x+2)(2-x)] - 2(x+2)/[(x+2)(2-x)]

5. Now, we can eliminate the denominators by multiplying both sides of the equation by [(x+2)(2-x)]:
   x^2 - 2x - x^2 - 4x - 4 = 2x - x^2 - 4x - 8

6. Simplify the equation:
   -6x - 4 = -2x - 8

7. Combine like terms:
   -6x + 2x = -8 + 4
   -4x = -4

8. Divide both sides by -4:
   x = 1

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